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

Percentage Accurate: 73.1% → 79.1%

Time: 14.9s

Alternatives: 25

Speedup: 1.0×

• 57.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), \frac{b}{y}, \mathsf{fma}\left(-i, j, \mathsf{fma}\left(z, x, \frac{\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a}{y}\right)\right)\right) \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -1.2e-84)
   (*
    (fma
     (fma (- z) c (* i t))
     (/ b y)
     (fma (- i) j (fma z x (/ (* (fma (- x) t (* j c)) a) y))))
    y)
   (if (<= y 2.8e+212)
     (fma
      (fma (- i) y (* c a))
      j
      (fma (fma (- x) a (* i b)) t (* (fma (- c) b (* y x)) z)))
     (* (fma (- i) j (* z x)) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -1.2e-84) {
		tmp = fma(fma(-z, c, (i * t)), (b / y), fma(-i, j, fma(z, x, ((fma(-x, t, (j * c)) * a) / y)))) * y;
	} else if (y <= 2.8e+212) {
		tmp = fma(fma(-i, y, (c * a)), j, fma(fma(-x, a, (i * b)), t, (fma(-c, b, (y * x)) * z)));
	} else {
		tmp = fma(-i, j, (z * x)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -1.2e-84)
		tmp = Float64(fma(fma(Float64(-z), c, Float64(i * t)), Float64(b / y), fma(Float64(-i), j, fma(z, x, Float64(Float64(fma(Float64(-x), t, Float64(j * c)) * a) / y)))) * y);
	elseif (y <= 2.8e+212)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(fma(Float64(-c), b, Float64(y * x)) * z)));
	else
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -1.2e-84], N[(N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * N[(b / y), $MachinePrecision] + N[((-i) * j + N[(z * x + N[(N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.8e+212], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.20000000000000009e-84

   1. Initial program 65.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 79.3%

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

   if -1.20000000000000009e-84 < y < 2.79999999999999997e212

   1. Initial program 77.7%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 85.1%

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

   if 2.79999999999999997e212 < y

   1. Initial program 50.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), \frac{b}{y}, \mathsf{fma}\left(-i, j, \mathsf{fma}\left(z, x, \frac{\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a}{y}\right)\right)\right) \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(c \cdot z - t \cdot i\right)\\ t_2 := j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - t\_1\right) + t\_2 \leq \infty:\\ \;\;\;\;\left(\frac{x}{{\left(\mathsf{fma}\left(-a, t, z \cdot y\right)\right)}^{-1}} - t\_1\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* c z) (* t i)))) (t_2 (* j (- (* c a) (* y i)))))
   (if (<= (+ (- (* x (- (* y z) (* t a))) t_1) t_2) INFINITY)
     (+ (- (/ x (pow (fma (- a) t (* z y)) -1.0)) t_1) t_2)
     (* (fma (- y) j (* b t)) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((c * z) - (t * i));
	double t_2 = j * ((c * a) - (y * i));
	double tmp;
	if ((((x * ((y * z) - (t * a))) - t_1) + t_2) <= ((double) INFINITY)) {
		tmp = ((x / pow(fma(-a, t, (z * y)), -1.0)) - t_1) + t_2;
	} else {
		tmp = fma(-y, j, (b * t)) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(c * z) - Float64(t * i)))
	t_2 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	tmp = 0.0
	if (Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - t_1) + t_2) <= Inf)
		tmp = Float64(Float64(Float64(x / (fma(Float64(-a), t, Float64(z * y)) ^ -1.0)) - t_1) + t_2);
	else
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], Infinity], N[(N[(N[(x / N[Power[N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 88.7

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 88.7

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \leq \infty:\\ \;\;\;\;\left(\frac{x}{{\left(\mathsf{fma}\left(-a, t, z \cdot y\right)\right)}^{-1}} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

4. Final simplification 81.3%

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

Alternative 4: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -7.2e+160)
   (fma (fma (- y) j (* b t)) i (* (fma (- t) a (* z y)) x))
   (fma
    (fma (- i) y (* c a))
    j
    (fma (fma (- x) a (* i b)) t (* (fma (- c) b (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -7.2e+160) {
		tmp = fma(fma(-y, j, (b * t)), i, (fma(-t, a, (z * y)) * x));
	} else {
		tmp = fma(fma(-i, y, (c * a)), j, fma(fma(-x, a, (i * b)), t, (fma(-c, b, (y * x)) * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -7.2e+160)
		tmp = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(fma(Float64(-c), b, Float64(y * x)) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -7.2e+160], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000042e160

   1. Initial program 73.9%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. cancel-sign-sub N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 81.7%

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

   if -7.20000000000000042e160 < x

   1. Initial program 70.9%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 80.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;\left(\left(\frac{y \cdot x}{c} - b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-301}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) t (* j c)) a)))
   (if (<= z -2.1e+118)
     (* (* (- (/ (* y x) c) b) c) z)
     (if (<= z -5e+33)
       (* (fma (- i) j (* z x)) y)
       (if (<= z -2.1e-173)
         (fma (- a) (* t x) (* (* i t) b))
         (if (<= z 6e-301)
           t_1
           (if (<= z 1.55e-109)
             (* (fma (- y) j (* b t)) i)
             (if (<= z 1.75e+139) t_1 (* (fma (- c) b (* y x)) z)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, t, (j * c)) * a;
	double tmp;
	if (z <= -2.1e+118) {
		tmp = ((((y * x) / c) - b) * c) * z;
	} else if (z <= -5e+33) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= -2.1e-173) {
		tmp = fma(-a, (t * x), ((i * t) * b));
	} else if (z <= 6e-301) {
		tmp = t_1;
	} else if (z <= 1.55e-109) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 1.75e+139) {
		tmp = t_1;
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), t, Float64(j * c)) * a)
	tmp = 0.0
	if (z <= -2.1e+118)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) / c) - b) * c) * z);
	elseif (z <= -5e+33)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= -2.1e-173)
		tmp = fma(Float64(-a), Float64(t * x), Float64(Float64(i * t) * b));
	elseif (z <= 6e-301)
		tmp = t_1;
	elseif (z <= 1.55e-109)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 1.75e+139)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[z, -2.1e+118], N[(N[(N[(N[(N[(y * x), $MachinePrecision] / c), $MachinePrecision] - b), $MachinePrecision] * c), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -5e+33], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -2.1e-173], N[((-a) * N[(t * x), $MachinePrecision] + N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-301], t$95$1, If[LessEqual[z, 1.55e-109], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.75e+139], t$95$1, N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -2.1e118

   1. Initial program 69.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 74.0%

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

   if -2.1e118 < z < -4.99999999999999973e33

   1. Initial program 67.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -4.99999999999999973e33 < z < -2.10000000000000001e-173

   1. Initial program 81.8%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. cancel-sign-sub N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.1%

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

   if -2.10000000000000001e-173 < z < 5.99999999999999998e-301 or 1.55e-109 < z < 1.74999999999999989e139

   1. Initial program 68.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 5.99999999999999998e-301 < z < 1.55e-109

   1. Initial program 72.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 1.74999999999999989e139 < z

   1. Initial program 71.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;\left(\left(\frac{y \cdot x}{c} - b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ t_2 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-301}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) t (* j c)) a)) (t_2 (* (fma (- c) b (* y x)) z)))
   (if (<= z -2.1e+118)
     t_2
     (if (<= z -5e+33)
       (* (fma (- i) j (* z x)) y)
       (if (<= z -2.1e-173)
         (fma (- a) (* t x) (* (* i t) b))
         (if (<= z 6e-301)
           t_1
           (if (<= z 1.55e-109)
             (* (fma (- y) j (* b t)) i)
             (if (<= z 1.75e+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 = fma(-x, t, (j * c)) * a;
	double t_2 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -2.1e+118) {
		tmp = t_2;
	} else if (z <= -5e+33) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= -2.1e-173) {
		tmp = fma(-a, (t * x), ((i * t) * b));
	} else if (z <= 6e-301) {
		tmp = t_1;
	} else if (z <= 1.55e-109) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 1.75e+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(fma(Float64(-x), t, Float64(j * c)) * a)
	t_2 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -2.1e+118)
		tmp = t_2;
	elseif (z <= -5e+33)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= -2.1e-173)
		tmp = fma(Float64(-a), Float64(t * x), Float64(Float64(i * t) * b));
	elseif (z <= 6e-301)
		tmp = t_1;
	elseif (z <= 1.55e-109)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 1.75e+139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.1e+118], t$95$2, If[LessEqual[z, -5e+33], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -2.1e-173], N[((-a) * N[(t * x), $MachinePrecision] + N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-301], t$95$1, If[LessEqual[z, 1.55e-109], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.75e+139], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.1e118 or 1.74999999999999989e139 < z

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -2.1e118 < z < -4.99999999999999973e33

   1. Initial program 67.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -4.99999999999999973e33 < z < -2.10000000000000001e-173

   1. Initial program 81.8%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. cancel-sign-sub N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.1%

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

   if -2.10000000000000001e-173 < z < 5.99999999999999998e-301 or 1.55e-109 < z < 1.74999999999999989e139

   1. Initial program 68.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if 5.99999999999999998e-301 < z < 1.55e-109

   1. Initial program 72.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \left(i \cdot t\right) \cdot b\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ t_2 := \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ t_3 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-294}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x))
        (t_2 (* (fma (- i) j (* z x)) y))
        (t_3 (* (fma (- c) b (* y x)) z)))
   (if (<= z -2.1e+118)
     t_3
     (if (<= z -1e+33)
       t_2
       (if (<= z -2.95e-294)
         t_1
         (if (<= z 5e-158)
           (* (* i t) b)
           (if (<= z 9.2e-38) t_2 (if (<= z 1.9e+139) t_1 t_3))))))))

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(-a, t, (z * y)) * x;
	double t_2 = fma(-i, j, (z * x)) * y;
	double t_3 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -2.1e+118) {
		tmp = t_3;
	} else if (z <= -1e+33) {
		tmp = t_2;
	} else if (z <= -2.95e-294) {
		tmp = t_1;
	} else if (z <= 5e-158) {
		tmp = (i * t) * b;
	} else if (z <= 9.2e-38) {
		tmp = t_2;
	} else if (z <= 1.9e+139) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	t_2 = Float64(fma(Float64(-i), j, Float64(z * x)) * y)
	t_3 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -2.1e+118)
		tmp = t_3;
	elseif (z <= -1e+33)
		tmp = t_2;
	elseif (z <= -2.95e-294)
		tmp = t_1;
	elseif (z <= 5e-158)
		tmp = Float64(Float64(i * t) * b);
	elseif (z <= 9.2e-38)
		tmp = t_2;
	elseif (z <= 1.9e+139)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.1e+118], t$95$3, If[LessEqual[z, -1e+33], t$95$2, If[LessEqual[z, -2.95e-294], t$95$1, If[LessEqual[z, 5e-158], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 9.2e-38], t$95$2, If[LessEqual[z, 1.9e+139], t$95$1, t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.1e118 or 1.9e139 < z

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -2.1e118 < z < -9.9999999999999995e32 or 4.99999999999999972e-158 < z < 9.20000000000000007e-38

   1. Initial program 70.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if -9.9999999999999995e32 < z < -2.94999999999999997e-294 or 9.20000000000000007e-38 < z < 1.9e139

   1. Initial program 73.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.9

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

   8. Applied rewrites 45.9%

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

   if -2.94999999999999997e-294 < z < 4.99999999999999972e-158

   1. Initial program 71.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 44.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ t_2 := \left(\left(-j\right) \cdot i\right) \cdot y\\ t_3 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+107}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-294}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-49}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x))
        (t_2 (* (* (- j) i) y))
        (t_3 (* (fma (- c) b (* y x)) z)))
   (if (<= z -9.5e+107)
     t_3
     (if (<= z -1.8e+44)
       t_2
       (if (<= z -2.95e-294)
         t_1
         (if (<= z 5.2e-158)
           (* (* i t) b)
           (if (<= z 4.7e-49) t_2 (if (<= z 1.9e+139) t_1 t_3))))))))

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(-a, t, (z * y)) * x;
	double t_2 = (-j * i) * y;
	double t_3 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -9.5e+107) {
		tmp = t_3;
	} else if (z <= -1.8e+44) {
		tmp = t_2;
	} else if (z <= -2.95e-294) {
		tmp = t_1;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 4.7e-49) {
		tmp = t_2;
	} else if (z <= 1.9e+139) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	t_2 = Float64(Float64(Float64(-j) * i) * y)
	t_3 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -9.5e+107)
		tmp = t_3;
	elseif (z <= -1.8e+44)
		tmp = t_2;
	elseif (z <= -2.95e-294)
		tmp = t_1;
	elseif (z <= 5.2e-158)
		tmp = Float64(Float64(i * t) * b);
	elseif (z <= 4.7e-49)
		tmp = t_2;
	elseif (z <= 1.9e+139)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-j) * i), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -9.5e+107], t$95$3, If[LessEqual[z, -1.8e+44], t$95$2, If[LessEqual[z, -2.95e-294], t$95$1, If[LessEqual[z, 5.2e-158], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 4.7e-49], t$95$2, If[LessEqual[z, 1.9e+139], t$95$1, t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.50000000000000019e107 or 1.9e139 < z

   1. Initial program 70.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -9.50000000000000019e107 < z < -1.8e44 or 5.2000000000000001e-158 < z < 4.70000000000000021e-49

   1. Initial program 69.1%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 62.0

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

   8. Applied rewrites 62.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.7%

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

   if -1.8e44 < z < -2.94999999999999997e-294 or 4.70000000000000021e-49 < z < 1.9e139

   1. Initial program 73.3%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -2.94999999999999997e-294 < z < 5.2000000000000001e-158

   1. Initial program 71.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 44.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-49}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(\frac{y \cdot x}{c} - b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -5.4e+221)
   (* (* (- (/ (* y x) c) b) c) z)
   (if (<= z 1.75e-57)
     (fma (fma (- x) a (* i b)) t (* (fma (- i) y (* c a)) j))
     (fma (fma (- t) x (* j c)) a (* (fma (- b) c (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -5.4e+221) {
		tmp = ((((y * x) / c) - b) * c) * z;
	} else if (z <= 1.75e-57) {
		tmp = fma(fma(-x, a, (i * b)), t, (fma(-i, y, (c * a)) * j));
	} else {
		tmp = fma(fma(-t, x, (j * c)), a, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -5.4e+221)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) / c) - b) * c) * z);
	elseif (z <= 1.75e-57)
		tmp = fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	else
		tmp = fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -5.4e+221], N[(N[(N[(N[(N[(y * x), $MachinePrecision] / c), $MachinePrecision] - b), $MachinePrecision] * c), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 1.75e-57], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.3999999999999999e221

   1. Initial program 73.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 95.1%

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

   if -5.3999999999999999e221 < z < 1.74999999999999996e-57

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. mul-1-neg N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 71.5%

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

   if 1.74999999999999996e-57 < z

   1. Initial program 67.2%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(\frac{y \cdot x}{c} - b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.5 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(-i\right) \cdot y\right) \cdot \left(j - \frac{b \cdot t}{y}\right)\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -2.5e+142)
   (* (* (- i) y) (- j (/ (* b t) y)))
   (if (<= i 1.05e+36)
     (fma (fma (- t) x (* j c)) a (* (fma (- b) c (* y x)) z))
     (* (fma (- y) j (* b t)) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2.5e+142) {
		tmp = (-i * y) * (j - ((b * t) / y));
	} else if (i <= 1.05e+36) {
		tmp = fma(fma(-t, x, (j * c)), a, (fma(-b, c, (y * x)) * z));
	} else {
		tmp = fma(-y, j, (b * t)) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -2.5e+142)
		tmp = Float64(Float64(Float64(-i) * y) * Float64(j - Float64(Float64(b * t) / y)));
	elseif (i <= 1.05e+36)
		tmp = fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	else
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -2.5e+142], N[(N[((-i) * y), $MachinePrecision] * N[(j - N[(N[(b * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.05e+36], N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.5000000000000001e142

   1. Initial program 72.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in i around -inf

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

   8. Applied rewrites 72.5%

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

   if -2.5000000000000001e142 < i < 1.05000000000000002e36

   1. Initial program 75.4%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 71.6%

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

   if 1.05000000000000002e36 < i

   1. Initial program 59.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.5 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(-i\right) \cdot y\right) \cdot \left(j - \frac{b \cdot t}{y}\right)\\ \mathbf{elif}\;i \leq 1.05 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ t_2 := \mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ t_3 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+118}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{-173}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-301}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) t (* j c)) a))
        (t_2 (* (fma (- y) j (* b t)) i))
        (t_3 (* (fma (- c) b (* y x)) z)))
   (if (<= z -3.2e+118)
     t_3
     (if (<= z -3.05e-173)
       t_2
       (if (<= z 6e-301)
         t_1
         (if (<= z 1.55e-109) t_2 (if (<= z 1.75e+139) t_1 t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, t, (j * c)) * a;
	double t_2 = fma(-y, j, (b * t)) * i;
	double t_3 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -3.2e+118) {
		tmp = t_3;
	} else if (z <= -3.05e-173) {
		tmp = t_2;
	} else if (z <= 6e-301) {
		tmp = t_1;
	} else if (z <= 1.55e-109) {
		tmp = t_2;
	} else if (z <= 1.75e+139) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), t, Float64(j * c)) * a)
	t_2 = Float64(fma(Float64(-y), j, Float64(b * t)) * i)
	t_3 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.2e+118)
		tmp = t_3;
	elseif (z <= -3.05e-173)
		tmp = t_2;
	elseif (z <= 6e-301)
		tmp = t_1;
	elseif (z <= 1.55e-109)
		tmp = t_2;
	elseif (z <= 1.75e+139)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.2e+118], t$95$3, If[LessEqual[z, -3.05e-173], t$95$2, If[LessEqual[z, 6e-301], t$95$1, If[LessEqual[z, 1.55e-109], t$95$2, If[LessEqual[z, 1.75e+139], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.20000000000000016e118 or 1.74999999999999989e139 < z

   1. Initial program 69.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -3.20000000000000016e118 < z < -3.0499999999999999e-173 or 5.99999999999999998e-301 < z < 1.55e-109

   1. Initial program 74.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   if -3.0499999999999999e-173 < z < 5.99999999999999998e-301 or 1.55e-109 < z < 1.74999999999999989e139

   1. Initial program 68.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+149} \lor \neg \left(t \leq 3.5 \cdot 10^{+131}\right):\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -2.9e+149) (not (<= t 3.5e+131)))
   (* (fma (- x) a (* i b)) t)
   (+ (* (* z x) y) (* j (- (* c a) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -2.9e+149) || !(t <= 3.5e+131)) {
		tmp = fma(-x, a, (i * b)) * t;
	} else {
		tmp = ((z * x) * y) + (j * ((c * a) - (y * i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -2.9e+149) || !(t <= 3.5e+131))
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	else
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(j * Float64(Float64(c * a) - Float64(y * i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -2.9e+149], N[Not[LessEqual[t, 3.5e+131]], $MachinePrecision]], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9000000000000002e149 or 3.4999999999999999e131 < t

   1. Initial program 55.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -2.9000000000000002e149 < t < 3.4999999999999999e131

   1. Initial program 77.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+149} \lor \neg \left(t \leq 3.5 \cdot 10^{+131}\right):\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- j) i) y)))
   (if (<= z -9.8e+107)
     (* (* x z) y)
     (if (<= z -3.2e+33)
       t_1
       (if (<= z 5.2e-158)
         (* (* i t) b)
         (if (<= z 2.5e-56)
           t_1
           (if (<= z 1.3e+77) (* (* a j) c) (* (* (- b) c) z))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-j * i) * y;
	double tmp;
	if (z <= -9.8e+107) {
		tmp = (x * z) * y;
	} else if (z <= -3.2e+33) {
		tmp = t_1;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.5e-56) {
		tmp = t_1;
	} else if (z <= 1.3e+77) {
		tmp = (a * j) * c;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-j * i) * y
    if (z <= (-9.8d+107)) then
        tmp = (x * z) * y
    else if (z <= (-3.2d+33)) then
        tmp = t_1
    else if (z <= 5.2d-158) then
        tmp = (i * t) * b
    else if (z <= 2.5d-56) then
        tmp = t_1
    else if (z <= 1.3d+77) then
        tmp = (a * j) * c
    else
        tmp = (-b * c) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-j * i) * y;
	double tmp;
	if (z <= -9.8e+107) {
		tmp = (x * z) * y;
	} else if (z <= -3.2e+33) {
		tmp = t_1;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.5e-56) {
		tmp = t_1;
	} else if (z <= 1.3e+77) {
		tmp = (a * j) * c;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-j * i) * y
	tmp = 0
	if z <= -9.8e+107:
		tmp = (x * z) * y
	elif z <= -3.2e+33:
		tmp = t_1
	elif z <= 5.2e-158:
		tmp = (i * t) * b
	elif z <= 2.5e-56:
		tmp = t_1
	elif z <= 1.3e+77:
		tmp = (a * j) * c
	else:
		tmp = (-b * c) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-j) * i) * y)
	tmp = 0.0
	if (z <= -9.8e+107)
		tmp = Float64(Float64(x * z) * y);
	elseif (z <= -3.2e+33)
		tmp = t_1;
	elseif (z <= 5.2e-158)
		tmp = Float64(Float64(i * t) * b);
	elseif (z <= 2.5e-56)
		tmp = t_1;
	elseif (z <= 1.3e+77)
		tmp = Float64(Float64(a * j) * c);
	else
		tmp = Float64(Float64(Float64(-b) * c) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-j * i) * y;
	tmp = 0.0;
	if (z <= -9.8e+107)
		tmp = (x * z) * y;
	elseif (z <= -3.2e+33)
		tmp = t_1;
	elseif (z <= 5.2e-158)
		tmp = (i * t) * b;
	elseif (z <= 2.5e-56)
		tmp = t_1;
	elseif (z <= 1.3e+77)
		tmp = (a * j) * c;
	else
		tmp = (-b * c) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-j) * i), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -9.8e+107], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -3.2e+33], t$95$1, If[LessEqual[z, 5.2e-158], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 2.5e-56], t$95$1, If[LessEqual[z, 1.3e+77], N[(N[(a * j), $MachinePrecision] * c), $MachinePrecision], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -9.8000000000000003e107

   1. Initial program 69.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 47.2

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 43.0%

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

   if -9.8000000000000003e107 < z < -3.20000000000000017e33 or 5.2000000000000001e-158 < z < 2.49999999999999999e-56

   1. Initial program 70.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate--l+ N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 61.1

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

   8. Applied rewrites 61.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.4%

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

   if -3.20000000000000017e33 < z < 5.2000000000000001e-158

   1. Initial program 76.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 38.6

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

   5. Applied rewrites 38.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 36.3%

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

   if 2.49999999999999999e-56 < z < 1.3000000000000001e77

   1. Initial program 68.3%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 43.0%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if 1.3000000000000001e77 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 72.0

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(b \cdot c\right)\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 50.6%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+175}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.25e+175)
   (* (* z y) x)
   (if (<= z 5.2e-158)
     (* (* i t) b)
     (if (<= z 2.6e-56)
       (* (* (- y) i) j)
       (if (<= z 1.3e+77) (* (* a j) c) (* (* (- b) c) z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.25e+175) {
		tmp = (z * y) * x;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.6e-56) {
		tmp = (-y * i) * j;
	} else if (z <= 1.3e+77) {
		tmp = (a * j) * c;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-2.25d+175)) then
        tmp = (z * y) * x
    else if (z <= 5.2d-158) then
        tmp = (i * t) * b
    else if (z <= 2.6d-56) then
        tmp = (-y * i) * j
    else if (z <= 1.3d+77) then
        tmp = (a * j) * c
    else
        tmp = (-b * c) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.25e+175) {
		tmp = (z * y) * x;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.6e-56) {
		tmp = (-y * i) * j;
	} else if (z <= 1.3e+77) {
		tmp = (a * j) * c;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.25e+175:
		tmp = (z * y) * x
	elif z <= 5.2e-158:
		tmp = (i * t) * b
	elif z <= 2.6e-56:
		tmp = (-y * i) * j
	elif z <= 1.3e+77:
		tmp = (a * j) * c
	else:
		tmp = (-b * c) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.25e+175)
		tmp = Float64(Float64(z * y) * x);
	elseif (z <= 5.2e-158)
		tmp = Float64(Float64(i * t) * b);
	elseif (z <= 2.6e-56)
		tmp = Float64(Float64(Float64(-y) * i) * j);
	elseif (z <= 1.3e+77)
		tmp = Float64(Float64(a * j) * c);
	else
		tmp = Float64(Float64(Float64(-b) * c) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -2.25e+175)
		tmp = (z * y) * x;
	elseif (z <= 5.2e-158)
		tmp = (i * t) * b;
	elseif (z <= 2.6e-56)
		tmp = (-y * i) * j;
	elseif (z <= 1.3e+77)
		tmp = (a * j) * c;
	else
		tmp = (-b * c) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.25e+175], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 5.2e-158], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 2.6e-56], N[(N[((-y) * i), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 1.3e+77], N[(N[(a * j), $MachinePrecision] * c), $MachinePrecision], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.24999999999999995e175

   1. Initial program 67.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 82.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.7%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if -2.24999999999999995e175 < z < 5.2000000000000001e-158

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 37.8

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 32.7%

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   if 5.2000000000000001e-158 < z < 2.59999999999999997e-56

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(b \cdot \left(i \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot \left(i \cdot t\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot \left(i \cdot t\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right) + \color{blue}{1} \cdot \left(b \cdot \left(i \cdot t\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right) + \color{blue}{b \cdot \left(i \cdot t\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + b \cdot \left(i \cdot t\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + b \cdot \color{blue}{\left(t \cdot i\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(b \cdot t\right) \cdot i}\right) \]

      8. cancel-sign-sub N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right)} \]

      9. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right) \]

      10. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right) \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i - \color{blue}{\left(-1 \cdot \left(b \cdot t\right)\right)} \cdot i\right) \]

      12. distribute-rgt-out-- N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)} \]

   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 45.0%

      \[\leadsto \left(-i\right) \cdot \color{blue}{\left(j \cdot y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.1%

       \[\leadsto \left(\left(-y\right) \cdot i\right) \cdot j \]

   if 2.59999999999999997e-56 < z < 1.3000000000000001e77

   1. Initial program 68.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 49.1

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 43.0%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if 1.3000000000000001e77 < z

   1. Initial program 66.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 72.0

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(b \cdot c\right)\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 50.6%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+175}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -2.1e+118)
     t_1
     (if (<= z -1.02e+35)
       (* (fma (- i) j (* z x)) y)
       (if (<= z 1.5e+139) (* (fma (- x) a (* i b)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -2.1e+118) {
		tmp = t_1;
	} else if (z <= -1.02e+35) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= 1.5e+139) {
		tmp = fma(-x, a, (i * b)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -2.1e+118)
		tmp = t_1;
	elseif (z <= -1.02e+35)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= 1.5e+139)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.1e+118], t$95$1, If[LessEqual[z, -1.02e+35], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.5e+139], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e118 or 1.5e139 < z

   1. Initial program 70.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 79.3

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   if -2.1e118 < z < -1.02000000000000007e35

   1. Initial program 71.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, j, x \cdot z\right)} \cdot y \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, j, x \cdot z\right) \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 65.9

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   if -1.02000000000000007e35 < z < 1.5e139

   1. Initial program 72.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot t \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \color{blue}{b \cdot i}\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

      12. lower-*.f64 50.7

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{if}\;j \leq -4.1 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;j \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) y (* c a)) j)))
   (if (<= j -4.1e+155)
     t_1
     (if (<= j 1.25e-18)
       (* (fma (- a) t (* z y)) x)
       (if (<= j 1.55e+77) (* (fma (- c) b (* 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 = fma(-i, y, (c * a)) * j;
	double tmp;
	if (j <= -4.1e+155) {
		tmp = t_1;
	} else if (j <= 1.25e-18) {
		tmp = fma(-a, t, (z * y)) * x;
	} else if (j <= 1.55e+77) {
		tmp = fma(-c, b, (y * x)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), y, Float64(c * a)) * j)
	tmp = 0.0
	if (j <= -4.1e+155)
		tmp = t_1;
	elseif (j <= 1.25e-18)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	elseif (j <= 1.55e+77)
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -4.1e+155], t$95$1, If[LessEqual[j, 1.25e-18], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[j, 1.55e+77], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -4.0999999999999998e155 or 1.54999999999999999e77 < j

   1. Initial program 64.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \cdot j \]

      3. mul-1-neg N/A

         \[\leadsto \left(a \cdot c + \color{blue}{-1 \cdot \left(i \cdot y\right)}\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right)} \cdot j \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right) \cdot j} \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot y} + a \cdot c\right) \cdot j \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)} \cdot j \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right) \cdot j \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

      11. lower-*.f64 66.3

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

   8. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j} \]

   if -4.0999999999999998e155 < j < 1.25000000000000009e-18

   1. Initial program 74.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. mul-1-neg N/A

         \[\leadsto \left(y \cdot z + \color{blue}{-1 \cdot \left(a \cdot t\right)}\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right)} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 51.0

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   8. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if 1.25000000000000009e-18 < j < 1.54999999999999999e77

   1. Initial program 81.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 59.4

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -4.1 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;j \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;j \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(j \cdot c\right) \cdot a\\ \mathbf{if}\;c \leq -6 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-57}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-263}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* j c) a)))
   (if (<= c -6e+128)
     t_1
     (if (<= c -4.8e-57)
       (* (* x z) y)
       (if (<= c 3.3e-263)
         (* (* (- t) a) x)
         (if (<= c 3.8e+18) (* (* z y) x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (j * c) * a;
	double tmp;
	if (c <= -6e+128) {
		tmp = t_1;
	} else if (c <= -4.8e-57) {
		tmp = (x * z) * y;
	} else if (c <= 3.3e-263) {
		tmp = (-t * a) * x;
	} else if (c <= 3.8e+18) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (j * c) * a
    if (c <= (-6d+128)) then
        tmp = t_1
    else if (c <= (-4.8d-57)) then
        tmp = (x * z) * y
    else if (c <= 3.3d-263) then
        tmp = (-t * a) * x
    else if (c <= 3.8d+18) then
        tmp = (z * y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (j * c) * a;
	double tmp;
	if (c <= -6e+128) {
		tmp = t_1;
	} else if (c <= -4.8e-57) {
		tmp = (x * z) * y;
	} else if (c <= 3.3e-263) {
		tmp = (-t * a) * x;
	} else if (c <= 3.8e+18) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (j * c) * a
	tmp = 0
	if c <= -6e+128:
		tmp = t_1
	elif c <= -4.8e-57:
		tmp = (x * z) * y
	elif c <= 3.3e-263:
		tmp = (-t * a) * x
	elif c <= 3.8e+18:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(j * c) * a)
	tmp = 0.0
	if (c <= -6e+128)
		tmp = t_1;
	elseif (c <= -4.8e-57)
		tmp = Float64(Float64(x * z) * y);
	elseif (c <= 3.3e-263)
		tmp = Float64(Float64(Float64(-t) * a) * x);
	elseif (c <= 3.8e+18)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (j * c) * a;
	tmp = 0.0;
	if (c <= -6e+128)
		tmp = t_1;
	elseif (c <= -4.8e-57)
		tmp = (x * z) * y;
	elseif (c <= 3.3e-263)
		tmp = (-t * a) * x;
	elseif (c <= 3.8e+18)
		tmp = (z * y) * x;
	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[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[c, -6e+128], t$95$1, If[LessEqual[c, -4.8e-57], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, 3.3e-263], N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[c, 3.8e+18], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.9999999999999997e128 or 3.8e18 < c

   1. Initial program 62.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot t\right)} + c \cdot j\right) \cdot a \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot t} + c \cdot j\right) \cdot a \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, t, c \cdot j\right)} \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, t, c \cdot j\right) \cdot a \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, c \cdot j\right) \cdot a \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{j \cdot c}\right) \cdot a \]

      9. lower-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{j \cdot c}\right) \cdot a \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 43.0%

      \[\leadsto \left(j \cdot c\right) \cdot a \]

   if -5.9999999999999997e128 < c < -4.80000000000000012e-57

   1. Initial program 65.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 53.5

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 46.0%

       \[\leadsto \left(x \cdot z\right) \cdot y \]

   if -4.80000000000000012e-57 < c < 3.2999999999999997e-263

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. mul-1-neg N/A

         \[\leadsto \left(y \cdot z + \color{blue}{-1 \cdot \left(a \cdot t\right)}\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right)} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 47.4

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   8. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot \left(a \cdot t\right)\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 38.2%

       \[\leadsto \left(\left(-t\right) \cdot a\right) \cdot x \]

   if 3.2999999999999997e-263 < c < 3.8e18

   1. Initial program 85.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 47.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{+128}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-57}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-263}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 27.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+175}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.25e+175)
   (* (* z y) x)
   (if (<= z 5.2e-158)
     (* (* i t) b)
     (if (<= z 2.6e-56)
       (* (* (- y) i) j)
       (if (<= z 4.5e+202) (* (* a j) c) (* (* x z) y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.25e+175) {
		tmp = (z * y) * x;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.6e-56) {
		tmp = (-y * i) * j;
	} else if (z <= 4.5e+202) {
		tmp = (a * j) * c;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-2.25d+175)) then
        tmp = (z * y) * x
    else if (z <= 5.2d-158) then
        tmp = (i * t) * b
    else if (z <= 2.6d-56) then
        tmp = (-y * i) * j
    else if (z <= 4.5d+202) then
        tmp = (a * j) * c
    else
        tmp = (x * z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.25e+175) {
		tmp = (z * y) * x;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.6e-56) {
		tmp = (-y * i) * j;
	} else if (z <= 4.5e+202) {
		tmp = (a * j) * c;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.25e+175:
		tmp = (z * y) * x
	elif z <= 5.2e-158:
		tmp = (i * t) * b
	elif z <= 2.6e-56:
		tmp = (-y * i) * j
	elif z <= 4.5e+202:
		tmp = (a * j) * c
	else:
		tmp = (x * z) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.25e+175)
		tmp = Float64(Float64(z * y) * x);
	elseif (z <= 5.2e-158)
		tmp = Float64(Float64(i * t) * b);
	elseif (z <= 2.6e-56)
		tmp = Float64(Float64(Float64(-y) * i) * j);
	elseif (z <= 4.5e+202)
		tmp = Float64(Float64(a * j) * c);
	else
		tmp = Float64(Float64(x * z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -2.25e+175)
		tmp = (z * y) * x;
	elseif (z <= 5.2e-158)
		tmp = (i * t) * b;
	elseif (z <= 2.6e-56)
		tmp = (-y * i) * j;
	elseif (z <= 4.5e+202)
		tmp = (a * j) * c;
	else
		tmp = (x * z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.25e+175], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 5.2e-158], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 2.6e-56], N[(N[((-y) * i), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 4.5e+202], N[(N[(a * j), $MachinePrecision] * c), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.24999999999999995e175

   1. Initial program 67.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 82.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.7%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if -2.24999999999999995e175 < z < 5.2000000000000001e-158

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 37.8

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 32.7%

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   if 5.2000000000000001e-158 < z < 2.59999999999999997e-56

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(b \cdot \left(i \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot \left(i \cdot t\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot \left(i \cdot t\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right) + \color{blue}{1} \cdot \left(b \cdot \left(i \cdot t\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right) + \color{blue}{b \cdot \left(i \cdot t\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + b \cdot \left(i \cdot t\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + b \cdot \color{blue}{\left(t \cdot i\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(b \cdot t\right) \cdot i}\right) \]

      8. cancel-sign-sub N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right)} \]

      9. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right) \]

      10. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right) \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i - \color{blue}{\left(-1 \cdot \left(b \cdot t\right)\right)} \cdot i\right) \]

      12. distribute-rgt-out-- N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)} \]

   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 45.0%

      \[\leadsto \left(-i\right) \cdot \color{blue}{\left(j \cdot y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.1%

       \[\leadsto \left(\left(-y\right) \cdot i\right) \cdot j \]

   if 2.59999999999999997e-56 < z < 4.49999999999999978e202

   1. Initial program 65.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 52.1

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 35.0%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if 4.49999999999999978e202 < z

   1. Initial program 69.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 62.5

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 58.7%

       \[\leadsto \left(x \cdot z\right) \cdot y \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+175}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+175}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.25e+175)
   (* (* z y) x)
   (if (<= z 5.2e-158)
     (* (* i t) b)
     (if (<= z 2.5e-56)
       (* (- i) (* j y))
       (if (<= z 4.5e+202) (* (* a j) c) (* (* x z) y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.25e+175) {
		tmp = (z * y) * x;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.5e-56) {
		tmp = -i * (j * y);
	} else if (z <= 4.5e+202) {
		tmp = (a * j) * c;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-2.25d+175)) then
        tmp = (z * y) * x
    else if (z <= 5.2d-158) then
        tmp = (i * t) * b
    else if (z <= 2.5d-56) then
        tmp = -i * (j * y)
    else if (z <= 4.5d+202) then
        tmp = (a * j) * c
    else
        tmp = (x * z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.25e+175) {
		tmp = (z * y) * x;
	} else if (z <= 5.2e-158) {
		tmp = (i * t) * b;
	} else if (z <= 2.5e-56) {
		tmp = -i * (j * y);
	} else if (z <= 4.5e+202) {
		tmp = (a * j) * c;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.25e+175:
		tmp = (z * y) * x
	elif z <= 5.2e-158:
		tmp = (i * t) * b
	elif z <= 2.5e-56:
		tmp = -i * (j * y)
	elif z <= 4.5e+202:
		tmp = (a * j) * c
	else:
		tmp = (x * z) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.25e+175)
		tmp = Float64(Float64(z * y) * x);
	elseif (z <= 5.2e-158)
		tmp = Float64(Float64(i * t) * b);
	elseif (z <= 2.5e-56)
		tmp = Float64(Float64(-i) * Float64(j * y));
	elseif (z <= 4.5e+202)
		tmp = Float64(Float64(a * j) * c);
	else
		tmp = Float64(Float64(x * z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -2.25e+175)
		tmp = (z * y) * x;
	elseif (z <= 5.2e-158)
		tmp = (i * t) * b;
	elseif (z <= 2.5e-56)
		tmp = -i * (j * y);
	elseif (z <= 4.5e+202)
		tmp = (a * j) * c;
	else
		tmp = (x * z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.25e+175], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 5.2e-158], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 2.5e-56], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+202], N[(N[(a * j), $MachinePrecision] * c), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.24999999999999995e175

   1. Initial program 67.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 82.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.7%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if -2.24999999999999995e175 < z < 5.2000000000000001e-158

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 37.8

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 32.7%

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   if 5.2000000000000001e-158 < z < 2.49999999999999999e-56

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(b \cdot \left(i \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot \left(i \cdot t\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot \left(i \cdot t\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right) + \color{blue}{1} \cdot \left(b \cdot \left(i \cdot t\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right) + \color{blue}{b \cdot \left(i \cdot t\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + b \cdot \left(i \cdot t\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + b \cdot \color{blue}{\left(t \cdot i\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \color{blue}{\left(b \cdot t\right) \cdot i}\right) \]

      8. cancel-sign-sub N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right)} \]

      9. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right) \]

      10. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} - \left(\mathsf{neg}\left(b \cdot t\right)\right) \cdot i\right) \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i - \color{blue}{\left(-1 \cdot \left(b \cdot t\right)\right)} \cdot i\right) \]

      12. distribute-rgt-out-- N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)} \]

   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 45.0%

      \[\leadsto \left(-i\right) \cdot \color{blue}{\left(j \cdot y\right)} \]

   if 2.49999999999999999e-56 < z < 4.49999999999999978e202

   1. Initial program 65.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 52.1

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 35.0%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if 4.49999999999999978e202 < z

   1. Initial program 69.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 62.5

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 58.7%

       \[\leadsto \left(x \cdot z\right) \cdot y \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+175}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.15 \cdot 10^{+156}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{elif}\;j \leq 2.1 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1.15e+156)
   (* (* a j) c)
   (if (<= j 2.1e+159) (* (fma (- a) t (* z y)) x) (* (* (- j) i) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1.15e+156) {
		tmp = (a * j) * c;
	} else if (j <= 2.1e+159) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = (-j * i) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1.15e+156)
		tmp = Float64(Float64(a * j) * c);
	elseif (j <= 2.1e+159)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = Float64(Float64(Float64(-j) * i) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1.15e+156], N[(N[(a * j), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[j, 2.1e+159], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[((-j) * i), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.1499999999999999e156

   1. Initial program 67.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 61.6

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if -1.1499999999999999e156 < j < 2.09999999999999989e159

   1. Initial program 73.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. mul-1-neg N/A

         \[\leadsto \left(y \cdot z + \color{blue}{-1 \cdot \left(a \cdot t\right)}\right) \cdot x \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right)} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 47.0

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   8. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if 2.09999999999999989e159 < j

   1. Initial program 62.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 49.4

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 52.3%

       \[\leadsto \left(\left(-j\right) \cdot i\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.15 \cdot 10^{+156}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{elif}\;j \leq 2.1 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 28.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot j\right) \cdot c\\ \mathbf{if}\;j \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 10:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;j \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* a j) c)))
   (if (<= j -7.5e+126)
     t_1
     (if (<= j 10.0) (* (* x z) y) (if (<= j 1.05e+144) (* (* i t) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * j) * c;
	double tmp;
	if (j <= -7.5e+126) {
		tmp = t_1;
	} else if (j <= 10.0) {
		tmp = (x * z) * y;
	} else if (j <= 1.05e+144) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * j) * c
    if (j <= (-7.5d+126)) then
        tmp = t_1
    else if (j <= 10.0d0) then
        tmp = (x * z) * y
    else if (j <= 1.05d+144) then
        tmp = (i * t) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * j) * c;
	double tmp;
	if (j <= -7.5e+126) {
		tmp = t_1;
	} else if (j <= 10.0) {
		tmp = (x * z) * y;
	} else if (j <= 1.05e+144) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (a * j) * c
	tmp = 0
	if j <= -7.5e+126:
		tmp = t_1
	elif j <= 10.0:
		tmp = (x * z) * y
	elif j <= 1.05e+144:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * j) * c)
	tmp = 0.0
	if (j <= -7.5e+126)
		tmp = t_1;
	elseif (j <= 10.0)
		tmp = Float64(Float64(x * z) * y);
	elseif (j <= 1.05e+144)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (a * j) * c;
	tmp = 0.0;
	if (j <= -7.5e+126)
		tmp = t_1;
	elseif (j <= 10.0)
		tmp = (x * z) * y;
	elseif (j <= 1.05e+144)
		tmp = (i * t) * b;
	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[(N[(a * j), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[j, -7.5e+126], t$95$1, If[LessEqual[j, 10.0], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[j, 1.05e+144], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -7.5000000000000006e126 or 1.04999999999999998e144 < j

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 55.6

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 47.4%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if -7.5000000000000006e126 < j < 10

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 38.0

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 32.3%

       \[\leadsto \left(x \cdot z\right) \cdot y \]

   if 10 < j < 1.04999999999999998e144

   1. Initial program 70.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 53.0

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 38.2%

      \[\leadsto \left(i \cdot t\right) \cdot b \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{elif}\;j \leq 10:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;j \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 28.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -7.5 \cdot 10^{+126} \lor \neg \left(j \leq 3.6 \cdot 10^{+141}\right):\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= j -7.5e+126) (not (<= j 3.6e+141))) (* (* a j) c) (* (* x z) y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -7.5e+126) || !(j <= 3.6e+141)) {
		tmp = (a * j) * c;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((j <= (-7.5d+126)) .or. (.not. (j <= 3.6d+141))) then
        tmp = (a * j) * c
    else
        tmp = (x * z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -7.5e+126) || !(j <= 3.6e+141)) {
		tmp = (a * j) * c;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (j <= -7.5e+126) or not (j <= 3.6e+141):
		tmp = (a * j) * c
	else:
		tmp = (x * z) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((j <= -7.5e+126) || !(j <= 3.6e+141))
		tmp = Float64(Float64(a * j) * c);
	else
		tmp = Float64(Float64(x * z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((j <= -7.5e+126) || ~((j <= 3.6e+141)))
		tmp = (a * j) * c;
	else
		tmp = (x * z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[j, -7.5e+126], N[Not[LessEqual[j, 3.6e+141]], $MachinePrecision]], N[(N[(a * j), $MachinePrecision] * c), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -7.5000000000000006e126 or 3.6000000000000001e141 < j

   1. Initial program 64.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 46.8%

      \[\leadsto \left(a \cdot j\right) \cdot c \]

   if -7.5000000000000006e126 < j < 3.6000000000000001e141

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 36.4

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 36.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 29.8%

       \[\leadsto \left(x \cdot z\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -7.5 \cdot 10^{+126} \lor \neg \left(j \leq 3.6 \cdot 10^{+141}\right):\\ \;\;\;\;\left(a \cdot j\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 28.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+115}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 15500000000000:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -2.85e+115)
   (* (* z y) x)
   (if (<= y 15500000000000.0) (* (* j c) a) (* (* x z) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -2.85e+115) {
		tmp = (z * y) * x;
	} else if (y <= 15500000000000.0) {
		tmp = (j * c) * a;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-2.85d+115)) then
        tmp = (z * y) * x
    else if (y <= 15500000000000.0d0) then
        tmp = (j * c) * a
    else
        tmp = (x * z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -2.85e+115) {
		tmp = (z * y) * x;
	} else if (y <= 15500000000000.0) {
		tmp = (j * c) * a;
	} else {
		tmp = (x * z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -2.85e+115:
		tmp = (z * y) * x
	elif y <= 15500000000000.0:
		tmp = (j * c) * a
	else:
		tmp = (x * z) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -2.85e+115)
		tmp = Float64(Float64(z * y) * x);
	elseif (y <= 15500000000000.0)
		tmp = Float64(Float64(j * c) * a);
	else
		tmp = Float64(Float64(x * z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -2.85e+115)
		tmp = (z * y) * x;
	elseif (y <= 15500000000000.0)
		tmp = (j * c) * a;
	else
		tmp = (x * z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -2.85e+115], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 15500000000000.0], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.84999999999999983e115

   1. Initial program 57.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 44.6

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if -2.84999999999999983e115 < y < 1.55e13

   1. Initial program 77.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot t\right)} + c \cdot j\right) \cdot a \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot t} + c \cdot j\right) \cdot a \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, t, c \cdot j\right)} \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, t, c \cdot j\right) \cdot a \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, c \cdot j\right) \cdot a \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{j \cdot c}\right) \cdot a \]

      9. lower-*.f64 46.7

         \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{j \cdot c}\right) \cdot a \]

   5. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 28.4%

      \[\leadsto \left(j \cdot c\right) \cdot a \]

   if 1.55e13 < y

   1. Initial program 64.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      6. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

      12. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      11. lower-*.f64 64.8

          \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 48.1%

       \[\leadsto \left(x \cdot z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+115}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 15500000000000:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 22.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot z\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* x z) y))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (x * z) * 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 * z) * 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 * z) * y;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (x * z) * y

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(x * z) * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (x * z) * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 71.4%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(\left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right)} \]

   4. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   6. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)}, j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, \left(t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\right) \]

   12. associate--l+ N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

5. Applied rewrites 76.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\right)\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

   3. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot i}\right)\right) + x \cdot z\right) \cdot y \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot i} + x \cdot z\right) \cdot y \]

   6. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot i + x \cdot z\right) \cdot y \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   11. lower-*.f64 39.7

       \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

8. Applied rewrites 39.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

9. Taylor expanded in x around inf

   \[\leadsto \left(x \cdot z\right) \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 24.0%

    \[\leadsto \left(x \cdot z\right) \cdot y \]

12. Final simplification 24.0%

    \[\leadsto \left(x \cdot z\right) \cdot y \]
13. Add Preprocessing

Alternative 25: 21.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z y) x))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (z * y) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * y) * x

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 71.4%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. *-commutative N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

   9. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   12. lower-*.f64 38.7

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 38.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 23.3%

   \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

9. Final simplification 23.3%

   \[\leadsto \left(z \cdot y\right) \cdot x \]
10. Add Preprocessing

Developer Target 1: 59.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024324 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))