Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 72.4% → 78.4%

Time: 18.3s

Alternatives: 28

Speedup: 0.9×

• 57.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(\mathsf{fma}\left(-b, z, j \cdot t\right), c, t\_1\right)\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+88}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x))
        (t_2 (fma (fma (- y) j (* b a)) i (fma (fma (- b) z (* j t)) c t_1))))
   (if (<= x -5.8e-26)
     t_2
     (if (<= x 2.75e-132)
       (fma
        (fma (- x) a (* j c))
        t
        (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b)))
       (if (<= x 4.2e+88) t_2 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(-a, t, (z * y)) * x;
	double t_2 = fma(fma(-y, j, (b * a)), i, fma(fma(-b, z, (j * t)), c, t_1));
	double tmp;
	if (x <= -5.8e-26) {
		tmp = t_2;
	} else if (x <= 2.75e-132) {
		tmp = fma(fma(-x, a, (j * c)), t, fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b)));
	} else if (x <= 4.2e+88) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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 = fma(fma(Float64(-y), j, Float64(b * a)), i, fma(fma(Float64(-b), z, Float64(j * t)), c, t_1))
	tmp = 0.0
	if (x <= -5.8e-26)
		tmp = t_2;
	elseif (x <= 2.75e-132)
		tmp = fma(fma(Float64(-x), a, Float64(j * c)), t, fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b)));
	elseif (x <= 4.2e+88)
		tmp = t_2;
	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[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-26], t$95$2, If[LessEqual[x, 2.75e-132], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+88], t$95$2, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999996e-26 or 2.75e-132 < x < 4.2e88

   1. Initial program 74.8%

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 46.3

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 85.4%

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

   if -5.7999999999999996e-26 < x < 2.75e-132

   1. Initial program 79.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 91.0%

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

   if 4.2e88 < x

   1. Initial program 71.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(\mathsf{fma}\left(-b, z, j \cdot t\right), c, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(\mathsf{fma}\left(-b, z, j \cdot t\right), c, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 91.7

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 91.7

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

   4. Applied rewrites 91.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

4. Final simplification 86.3%

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

Alternative 3: 81.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

4. Final simplification 86.2%

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

Alternative 4: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, t\_1\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b))))
   (if (<= y -1.85e-7)
     (fma (fma (- x) a (* j c)) t t_1)
     (if (<= y 3.6e+170)
       (fma
        (fma (- y) j (* b a))
        i
        (fma (fma (- b) c (* y x)) z (* (fma (- a) x (* j c)) t)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b));
	double tmp;
	if (y <= -1.85e-7) {
		tmp = fma(fma(-x, a, (j * c)), t, t_1);
	} else if (y <= 3.6e+170) {
		tmp = fma(fma(-y, j, (b * a)), i, fma(fma(-b, c, (y * x)), z, (fma(-a, x, (j * c)) * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b))
	tmp = 0.0
	if (y <= -1.85e-7)
		tmp = fma(fma(Float64(-x), a, Float64(j * c)), t, t_1);
	elseif (y <= 3.6e+170)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-a), x, Float64(j * c)) * 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[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.85e-7], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t + t$95$1), $MachinePrecision], If[LessEqual[y, 3.6e+170], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85000000000000002e-7

   1. Initial program 75.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 87.6%

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

   if -1.85000000000000002e-7 < y < 3.6e170

   1. Initial program 80.4%

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 45.3

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

   7. Applied rewrites 45.3%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 82.2%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 81.6%

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

   if 3.6e170 < y

   1. Initial program 57.9%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 85.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))))
   (if (<= z -3.4e+51)
     t_1
     (if (<= z 5.4e+230)
       (fma
        (fma (- x) a (* j c))
        t
        (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	double tmp;
	if (z <= -3.4e+51) {
		tmp = t_1;
	} else if (z <= 5.4e+230) {
		tmp = fma(fma(-x, a, (j * c)), t, fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j))
	tmp = 0.0
	if (z <= -3.4e+51)
		tmp = t_1;
	elseif (z <= 5.4e+230)
		tmp = fma(fma(Float64(-x), a, Float64(j * c)), t, fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e+51], t$95$1, If[LessEqual[z, 5.4e+230], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.39999999999999984e51 or 5.40000000000000006e230 < z

   1. Initial program 67.1%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 77.8%

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

   if -3.39999999999999984e51 < z < 5.40000000000000006e230

   1. Initial program 79.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 83.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, j \cdot c\right), t, \mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ t_2 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-285}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \left(\left(-c\right) \cdot z\right) \cdot b\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j)))
        (t_2 (* (fma (- a) t (* z y)) x)))
   (if (<= x -7.4e+95)
     t_2
     (if (<= x -1.46e-285)
       t_1
       (if (<= x 4.5e-50)
         (fma (fma (- y) j (* b a)) i (* (* (- c) z) b))
         (if (<= x 2.5e+20) 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(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	double t_2 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -7.4e+95) {
		tmp = t_2;
	} else if (x <= -1.46e-285) {
		tmp = t_1;
	} else if (x <= 4.5e-50) {
		tmp = fma(fma(-y, j, (b * a)), i, ((-c * z) * b));
	} else if (x <= 2.5e+20) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j))
	t_2 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -7.4e+95)
		tmp = t_2;
	elseif (x <= -1.46e-285)
		tmp = t_1;
	elseif (x <= 4.5e-50)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(Float64(Float64(-c) * z) * b));
	elseif (x <= 2.5e+20)
		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[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -7.4e+95], t$95$2, If[LessEqual[x, -1.46e-285], t$95$1, If[LessEqual[x, 4.5e-50], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e+20], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4000000000000003e95 or 2.5e20 < x

   1. Initial program 74.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -7.4000000000000003e95 < x < -1.45999999999999992e-285 or 4.49999999999999962e-50 < x < 2.5e20

   1. Initial program 73.8%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   if -1.45999999999999992e-285 < x < 4.49999999999999962e-50

   1. Initial program 82.2%

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 79.5%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 75.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \left(\left(-c\right) \cdot z\right) \cdot b\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -7.4e+95)
     t_1
     (if (<= x -1.75e-285)
       (fma (fma (- b) c (* y x)) z (* (fma (- i) y (* c t)) j))
       (if (<= x 1.7e+35)
         (fma (fma (- j) i (* z x)) y (* (fma (- c) z (* i a)) b))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -7.4e+95) {
		tmp = t_1;
	} else if (x <= -1.75e-285) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-i, y, (c * t)) * j));
	} else if (x <= 1.7e+35) {
		tmp = fma(fma(-j, i, (z * x)), y, (fma(-c, z, (i * a)) * b));
	} else {
		tmp = t_1;
	}
	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)
	tmp = 0.0
	if (x <= -7.4e+95)
		tmp = t_1;
	elseif (x <= -1.75e-285)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	elseif (x <= 1.7e+35)
		tmp = fma(fma(Float64(-j), i, Float64(z * x)), y, Float64(fma(Float64(-c), z, Float64(i * a)) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -7.4e+95], t$95$1, If[LessEqual[x, -1.75e-285], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7e+35], N[(N[((-j) * i + N[(z * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4000000000000003e95 or 1.7000000000000001e35 < x

   1. Initial program 75.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -7.4000000000000003e95 < x < -1.7500000000000002e-285

   1. Initial program 75.9%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 71.8%

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

   if -1.7500000000000002e-285 < x < 1.7000000000000001e35

   1. Initial program 77.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

   5. Applied rewrites 76.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, i, z \cdot x\right), y, \mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-c\right) \cdot z\right) \cdot b\\ t_2 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+94}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-204}:\\ \;\;\;\;t\_1 + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- c) z) b)) (t_2 (* (fma (- a) t (* z y)) x)))
   (if (<= x -2.2e+94)
     t_2
     (if (<= x -4.4e-204)
       (+ t_1 (* (- (* c t) (* i y)) j))
       (if (<= x 1.1e+35) (fma (fma (- y) j (* b a)) i 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 = (-c * z) * b;
	double t_2 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -2.2e+94) {
		tmp = t_2;
	} else if (x <= -4.4e-204) {
		tmp = t_1 + (((c * t) - (i * y)) * j);
	} else if (x <= 1.1e+35) {
		tmp = fma(fma(-y, j, (b * a)), i, t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-c) * z) * b)
	t_2 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -2.2e+94)
		tmp = t_2;
	elseif (x <= -4.4e-204)
		tmp = Float64(t_1 + Float64(Float64(Float64(c * t) - Float64(i * y)) * j));
	elseif (x <= 1.1e+35)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, 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[((-c) * z), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.2e+94], t$95$2, If[LessEqual[x, -4.4e-204], N[(t$95$1 + N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+35], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.20000000000000012e94 or 1.0999999999999999e35 < x

   1. Initial program 75.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -2.20000000000000012e94 < x < -4.3999999999999997e-204

   1. Initial program 75.3%

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

   3. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if -4.3999999999999997e-204 < x < 1.0999999999999999e35

   1. Initial program 77.3%

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 80.5%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 70.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \left(\left(-c\right) \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-259}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \left(\left(-c\right) \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -2.4e+94)
     t_1
     (if (<= x -1.28e-259)
       (+ (* (* i b) a) (* (- (* c t) (* i y)) j))
       (if (<= x 1.1e+35)
         (fma (fma (- y) j (* b a)) i (* (* (- c) z) b))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -2.4e+94) {
		tmp = t_1;
	} else if (x <= -1.28e-259) {
		tmp = ((i * b) * a) + (((c * t) - (i * y)) * j);
	} else if (x <= 1.1e+35) {
		tmp = fma(fma(-y, j, (b * a)), i, ((-c * z) * b));
	} else {
		tmp = t_1;
	}
	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)
	tmp = 0.0
	if (x <= -2.4e+94)
		tmp = t_1;
	elseif (x <= -1.28e-259)
		tmp = Float64(Float64(Float64(i * b) * a) + Float64(Float64(Float64(c * t) - Float64(i * y)) * j));
	elseif (x <= 1.1e+35)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(Float64(Float64(-c) * z) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.4e+94], t$95$1, If[LessEqual[x, -1.28e-259], N[(N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+35], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999983e94 or 1.0999999999999999e35 < x

   1. Initial program 75.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -2.39999999999999983e94 < x < -1.27999999999999998e-259

   1. Initial program 76.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -1.27999999999999998e-259 < x < 1.0999999999999999e35

   1. Initial program 76.2%

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 40.2

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

   7. Applied rewrites 40.2%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 79.9%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 71.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-259}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \left(\left(-c\right) \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, i \cdot b\right)\\ t_2 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.75 \cdot 10^{+93}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 33000000000:\\ \;\;\;\;\mathsf{fma}\left(t\_1, a, \left(j \cdot t\right) \cdot c\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, a, \left(\left(-i\right) \cdot y\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- x) t (* i b))) (t_2 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.75e+93)
     t_2
     (if (<= z 33000000000.0)
       (fma t_1 a (* (* j t) c))
       (if (<= z 9.2e+97) (fma t_1 a (* (* (- i) y) j)) 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, (i * b));
	double t_2 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.75e+93) {
		tmp = t_2;
	} else if (z <= 33000000000.0) {
		tmp = fma(t_1, a, ((j * t) * c));
	} else if (z <= 9.2e+97) {
		tmp = fma(t_1, a, ((-i * y) * j));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-x), t, Float64(i * b))
	t_2 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.75e+93)
		tmp = t_2;
	elseif (z <= 33000000000.0)
		tmp = fma(t_1, a, Float64(Float64(j * t) * c));
	elseif (z <= 9.2e+97)
		tmp = fma(t_1, a, Float64(Float64(Float64(-i) * y) * j));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.75e+93], t$95$2, If[LessEqual[z, 33000000000.0], N[(t$95$1 * a + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.2e+97], N[(t$95$1 * a + N[(N[((-i) * y), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.7500000000000001e93 or 9.20000000000000022e97 < z

   1. Initial program 66.7%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -3.7500000000000001e93 < z < 3.3e10

   1. Initial program 82.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 63.5%

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

   if 3.3e10 < z < 9.20000000000000022e97

   1. Initial program 76.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 53.8%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 82.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.75 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 33000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \left(j \cdot t\right) \cdot c\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \left(\left(-i\right) \cdot y\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, t\_1\right)\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{-67}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) y (* c t)) j))
        (t_2 (fma (fma (- b) c (* y x)) z t_1)))
   (if (<= z -4.1e-67)
     t_2
     (if (<= z 1.1e+98) (fma (fma (- x) t (* i b)) a 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(-i, y, (c * t)) * j;
	double t_2 = fma(fma(-b, c, (y * x)), z, t_1);
	double tmp;
	if (z <= -4.1e-67) {
		tmp = t_2;
	} else if (z <= 1.1e+98) {
		tmp = fma(fma(-x, t, (i * b)), a, 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(-i), y, Float64(c * t)) * j)
	t_2 = fma(fma(Float64(-b), c, Float64(y * x)), z, t_1)
	tmp = 0.0
	if (z <= -4.1e-67)
		tmp = t_2;
	elseif (z <= 1.1e+98)
		tmp = fma(fma(Float64(-x), t, Float64(i * b)), a, 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[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + t$95$1), $MachinePrecision]}, If[LessEqual[z, -4.1e-67], t$95$2, If[LessEqual[z, 1.1e+98], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a + t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0999999999999997e-67 or 1.10000000000000004e98 < z

   1. Initial program 69.3%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 73.6%

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

   if -4.0999999999999997e-67 < z < 1.10000000000000004e98

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 74.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \left(\left(-c\right) \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -5.1e+95)
     t_1
     (if (<= x 1.1e+35) (fma (fma (- y) j (* b a)) i (* (* (- c) z) b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -5.1e+95) {
		tmp = t_1;
	} else if (x <= 1.1e+35) {
		tmp = fma(fma(-y, j, (b * a)), i, ((-c * z) * b));
	} else {
		tmp = t_1;
	}
	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)
	tmp = 0.0
	if (x <= -5.1e+95)
		tmp = t_1;
	elseif (x <= 1.1e+35)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(Float64(Float64(-c) * z) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.1e+95], t$95$1, If[LessEqual[x, 1.1e+35], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.10000000000000003e95 or 1.0999999999999999e35 < x

   1. Initial program 75.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -5.10000000000000003e95 < x < 1.0999999999999999e35

   1. Initial program 76.5%

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 36.6

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

   7. Applied rewrites 36.6%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 76.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 62.0%

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

Alternative 13: 29.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+212}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- t) x) a)))
   (if (<= x -1.8e+212)
     (* (* y x) z)
     (if (<= x -4.3e-63)
       t_1
       (if (<= x -6.8e-224)
         (* (* (- c) z) b)
         (if (<= x 2.6e-20)
           (* (* i b) a)
           (if (<= x 2.1e+31) (* (* (- j) y) i) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-t * x) * a;
	double tmp;
	if (x <= -1.8e+212) {
		tmp = (y * x) * z;
	} else if (x <= -4.3e-63) {
		tmp = t_1;
	} else if (x <= -6.8e-224) {
		tmp = (-c * z) * b;
	} else if (x <= 2.6e-20) {
		tmp = (i * b) * a;
	} else if (x <= 2.1e+31) {
		tmp = (-j * y) * i;
	} 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 = (-t * x) * a
    if (x <= (-1.8d+212)) then
        tmp = (y * x) * z
    else if (x <= (-4.3d-63)) then
        tmp = t_1
    else if (x <= (-6.8d-224)) then
        tmp = (-c * z) * b
    else if (x <= 2.6d-20) then
        tmp = (i * b) * a
    else if (x <= 2.1d+31) then
        tmp = (-j * y) * i
    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 = (-t * x) * a;
	double tmp;
	if (x <= -1.8e+212) {
		tmp = (y * x) * z;
	} else if (x <= -4.3e-63) {
		tmp = t_1;
	} else if (x <= -6.8e-224) {
		tmp = (-c * z) * b;
	} else if (x <= 2.6e-20) {
		tmp = (i * b) * a;
	} else if (x <= 2.1e+31) {
		tmp = (-j * y) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-t * x) * a
	tmp = 0
	if x <= -1.8e+212:
		tmp = (y * x) * z
	elif x <= -4.3e-63:
		tmp = t_1
	elif x <= -6.8e-224:
		tmp = (-c * z) * b
	elif x <= 2.6e-20:
		tmp = (i * b) * a
	elif x <= 2.1e+31:
		tmp = (-j * y) * i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-t) * x) * a)
	tmp = 0.0
	if (x <= -1.8e+212)
		tmp = Float64(Float64(y * x) * z);
	elseif (x <= -4.3e-63)
		tmp = t_1;
	elseif (x <= -6.8e-224)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (x <= 2.6e-20)
		tmp = Float64(Float64(i * b) * a);
	elseif (x <= 2.1e+31)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-t * x) * a;
	tmp = 0.0;
	if (x <= -1.8e+212)
		tmp = (y * x) * z;
	elseif (x <= -4.3e-63)
		tmp = t_1;
	elseif (x <= -6.8e-224)
		tmp = (-c * z) * b;
	elseif (x <= 2.6e-20)
		tmp = (i * b) * a;
	elseif (x <= 2.1e+31)
		tmp = (-j * y) * i;
	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[((-t) * x), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[x, -1.8e+212], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, -4.3e-63], t$95$1, If[LessEqual[x, -6.8e-224], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 2.6e-20], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 2.1e+31], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.8e212

   1. Initial program 75.5%

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 51.1

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

   7. Applied rewrites 51.1%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-*.f64 66.3

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

   10. Applied rewrites 66.3%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 61.5%

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

   if -1.8e212 < x < -4.2999999999999999e-63 or 2.09999999999999979e31 < x

   1. Initial program 77.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.5%

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

   if -4.2999999999999999e-63 < x < -6.79999999999999984e-224

   1. Initial program 68.5%

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 21.0

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

   7. Applied rewrites 21.0%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-*.f64 41.1

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

   10. Applied rewrites 41.1%

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

   11. Taylor expanded in c around inf

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

   13. Applied rewrites 34.3%

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

   if -6.79999999999999984e-224 < x < 2.59999999999999995e-20

   1. Initial program 83.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 41.1%

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

   if 2.59999999999999995e-20 < x < 2.09999999999999979e31

   1. Initial program 54.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 45.9%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+212}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-63}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ t_2 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-287}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) z (* j t)) c)) (t_2 (* (fma (- a) t (* z y)) x)))
   (if (<= x -2.2e+82)
     t_2
     (if (<= x -5.2e-287)
       t_1
       (if (<= x 6.5e-161) (* (* i b) a) (if (<= x 2.25e+20) 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(-b, z, (j * t)) * c;
	double t_2 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -2.2e+82) {
		tmp = t_2;
	} else if (x <= -5.2e-287) {
		tmp = t_1;
	} else if (x <= 6.5e-161) {
		tmp = (i * b) * a;
	} else if (x <= 2.25e+20) {
		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(-b), z, Float64(j * t)) * c)
	t_2 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -2.2e+82)
		tmp = t_2;
	elseif (x <= -5.2e-287)
		tmp = t_1;
	elseif (x <= 6.5e-161)
		tmp = Float64(Float64(i * b) * a);
	elseif (x <= 2.25e+20)
		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[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.2e+82], t$95$2, If[LessEqual[x, -5.2e-287], t$95$1, If[LessEqual[x, 6.5e-161], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 2.25e+20], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2000000000000001e82 or 2.25e20 < x

   1. Initial program 74.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.2000000000000001e82 < x < -5.1999999999999999e-287 or 6.50000000000000008e-161 < x < 2.25e20

   1. Initial program 74.9%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   if -5.1999999999999999e-287 < x < 6.50000000000000008e-161

   1. Initial program 84.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 53.1%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ t_2 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 40:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)) (t_2 (* (fma (- a) t (* z y)) x)))
   (if (<= x -2.9e+85)
     t_2
     (if (<= x -6.2e-224)
       t_1
       (if (<= x 4.5e-94) (* (* i b) a) (if (<= x 40.0) 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(-b, c, (y * x)) * z;
	double t_2 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -2.9e+85) {
		tmp = t_2;
	} else if (x <= -6.2e-224) {
		tmp = t_1;
	} else if (x <= 4.5e-94) {
		tmp = (i * b) * a;
	} else if (x <= 40.0) {
		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(-b), c, Float64(y * x)) * z)
	t_2 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -2.9e+85)
		tmp = t_2;
	elseif (x <= -6.2e-224)
		tmp = t_1;
	elseif (x <= 4.5e-94)
		tmp = Float64(Float64(i * b) * a);
	elseif (x <= 40.0)
		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[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.9e+85], t$95$2, If[LessEqual[x, -6.2e-224], t$95$1, If[LessEqual[x, 4.5e-94], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 40.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.89999999999999997e85 or 40 < x

   1. Initial program 72.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -2.89999999999999997e85 < x < -6.20000000000000017e-224 or 4.5000000000000002e-94 < x < 40

   1. Initial program 75.8%

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

   3. Taylor expanded in z around inf

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

      1. *-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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 42.0

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

   5. Applied rewrites 42.0%

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

   if -6.20000000000000017e-224 < x < 4.5000000000000002e-94

   1. Initial program 82.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 42.7%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 40:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.75 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \left(j \cdot t\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -3.75e+93)
     t_1
     (if (<= z 8.2e+80) (fma (fma (- x) t (* i b)) a (* (* j t) c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -3.75e+93) {
		tmp = t_1;
	} else if (z <= 8.2e+80) {
		tmp = fma(fma(-x, t, (i * b)), a, ((j * t) * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3.75e+93)
		tmp = t_1;
	elseif (z <= 8.2e+80)
		tmp = fma(fma(Float64(-x), t, Float64(i * b)), a, Float64(Float64(j * t) * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.75e+93], t$95$1, If[LessEqual[z, 8.2e+80], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7500000000000001e93 or 8.20000000000000003e80 < z

   1. Initial program 67.7%

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

   3. Taylor expanded in z around inf

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

      1. *-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. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -3.7500000000000001e93 < z < 8.20000000000000003e80

   1. Initial program 81.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-lft-in N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 62.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.75 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \left(j \cdot t\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-j, t, b \cdot z\right) \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -1.6e+94)
     t_1
     (if (<= x 2e-160)
       (* (fma (- c) z (* i a)) b)
       (if (<= x 2.25e+20) (* (fma (- j) t (* b z)) (- c)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -1.6e+94) {
		tmp = t_1;
	} else if (x <= 2e-160) {
		tmp = fma(-c, z, (i * a)) * b;
	} else if (x <= 2.25e+20) {
		tmp = fma(-j, t, (b * z)) * -c;
	} else {
		tmp = t_1;
	}
	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)
	tmp = 0.0
	if (x <= -1.6e+94)
		tmp = t_1;
	elseif (x <= 2e-160)
		tmp = Float64(fma(Float64(-c), z, Float64(i * a)) * b);
	elseif (x <= 2.25e+20)
		tmp = Float64(fma(Float64(-j), t, Float64(b * z)) * Float64(-c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.6e+94], t$95$1, If[LessEqual[x, 2e-160], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 2.25e+20], N[(N[((-j) * t + N[(b * z), $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000007e94 or 2.25e20 < x

   1. Initial program 74.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -1.60000000000000007e94 < x < 2e-160

   1. Initial program 78.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if 2e-160 < x < 2.25e20

   1. Initial program 73.7%

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in c around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.1

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

   7. Applied rewrites 58.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-j, t, b \cdot z\right) \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(-c\right) \cdot b\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- a) t) x)))
   (if (<= t -2.1e+50)
     t_1
     (if (<= t -3.3e+21)
       (* (* (- i) j) y)
       (if (<= t -1e-298)
         (* (* y x) z)
         (if (<= t 3.6e+102) (* (* (- c) b) 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 = (-a * t) * x;
	double tmp;
	if (t <= -2.1e+50) {
		tmp = t_1;
	} else if (t <= -3.3e+21) {
		tmp = (-i * j) * y;
	} else if (t <= -1e-298) {
		tmp = (y * x) * z;
	} else if (t <= 3.6e+102) {
		tmp = (-c * b) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-a * t) * x
    if (t <= (-2.1d+50)) then
        tmp = t_1
    else if (t <= (-3.3d+21)) then
        tmp = (-i * j) * y
    else if (t <= (-1d-298)) then
        tmp = (y * x) * z
    else if (t <= 3.6d+102) then
        tmp = (-c * b) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-a * t) * x;
	double tmp;
	if (t <= -2.1e+50) {
		tmp = t_1;
	} else if (t <= -3.3e+21) {
		tmp = (-i * j) * y;
	} else if (t <= -1e-298) {
		tmp = (y * x) * z;
	} else if (t <= 3.6e+102) {
		tmp = (-c * b) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-a * t) * x
	tmp = 0
	if t <= -2.1e+50:
		tmp = t_1
	elif t <= -3.3e+21:
		tmp = (-i * j) * y
	elif t <= -1e-298:
		tmp = (y * x) * z
	elif t <= 3.6e+102:
		tmp = (-c * b) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-a) * t) * x)
	tmp = 0.0
	if (t <= -2.1e+50)
		tmp = t_1;
	elseif (t <= -3.3e+21)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (t <= -1e-298)
		tmp = Float64(Float64(y * x) * z);
	elseif (t <= 3.6e+102)
		tmp = Float64(Float64(Float64(-c) * b) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-a * t) * x;
	tmp = 0.0;
	if (t <= -2.1e+50)
		tmp = t_1;
	elseif (t <= -3.3e+21)
		tmp = (-i * j) * y;
	elseif (t <= -1e-298)
		tmp = (y * x) * z;
	elseif (t <= 3.6e+102)
		tmp = (-c * b) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -2.1e+50], t$95$1, If[LessEqual[t, -3.3e+21], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, -1e-298], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 3.6e+102], N[(N[((-c) * b), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.1e50 or 3.6000000000000002e102 < t

   1. Initial program 65.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 43.9%

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

   if -2.1e50 < t < -3.3e21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.7%

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

   if -3.3e21 < t < -9.99999999999999912e-299

   1. Initial program 81.1%

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 28.7

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 52.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 52.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around 0

       \[\leadsto \left(x \cdot y\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 37.9%

       \[\leadsto \left(x \cdot y\right) \cdot z \]

   if -9.99999999999999912e-299 < t < 3.6000000000000002e102

   1. Initial program 82.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   4. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot t\right), j, \mathsf{fma}\left(-b, \mathsf{fma}\left(-a, i, c \cdot z\right), \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 33.8

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 51.8

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 51.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around inf

       \[\leadsto \left(-1 \cdot \left(b \cdot c\right)\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 35.0%

       \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+50}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(-c\right) \cdot b\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+212}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- t) x) a)))
   (if (<= x -1.8e+212)
     (* (* y x) z)
     (if (<= x -1.6e+94)
       t_1
       (if (<= x 2.6e-20)
         (* (* i b) a)
         (if (<= x 2.1e+31) (* (* (- j) y) i) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-t * x) * a;
	double tmp;
	if (x <= -1.8e+212) {
		tmp = (y * x) * z;
	} else if (x <= -1.6e+94) {
		tmp = t_1;
	} else if (x <= 2.6e-20) {
		tmp = (i * b) * a;
	} else if (x <= 2.1e+31) {
		tmp = (-j * y) * i;
	} 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 = (-t * x) * a
    if (x <= (-1.8d+212)) then
        tmp = (y * x) * z
    else if (x <= (-1.6d+94)) then
        tmp = t_1
    else if (x <= 2.6d-20) then
        tmp = (i * b) * a
    else if (x <= 2.1d+31) then
        tmp = (-j * y) * i
    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 = (-t * x) * a;
	double tmp;
	if (x <= -1.8e+212) {
		tmp = (y * x) * z;
	} else if (x <= -1.6e+94) {
		tmp = t_1;
	} else if (x <= 2.6e-20) {
		tmp = (i * b) * a;
	} else if (x <= 2.1e+31) {
		tmp = (-j * y) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-t * x) * a
	tmp = 0
	if x <= -1.8e+212:
		tmp = (y * x) * z
	elif x <= -1.6e+94:
		tmp = t_1
	elif x <= 2.6e-20:
		tmp = (i * b) * a
	elif x <= 2.1e+31:
		tmp = (-j * y) * i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-t) * x) * a)
	tmp = 0.0
	if (x <= -1.8e+212)
		tmp = Float64(Float64(y * x) * z);
	elseif (x <= -1.6e+94)
		tmp = t_1;
	elseif (x <= 2.6e-20)
		tmp = Float64(Float64(i * b) * a);
	elseif (x <= 2.1e+31)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-t * x) * a;
	tmp = 0.0;
	if (x <= -1.8e+212)
		tmp = (y * x) * z;
	elseif (x <= -1.6e+94)
		tmp = t_1;
	elseif (x <= 2.6e-20)
		tmp = (i * b) * a;
	elseif (x <= 2.1e+31)
		tmp = (-j * y) * i;
	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[((-t) * x), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[x, -1.8e+212], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, -1.6e+94], t$95$1, If[LessEqual[x, 2.6e-20], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 2.1e+31], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.8e212

   1. Initial program 75.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   4. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot t\right), j, \mathsf{fma}\left(-b, \mathsf{fma}\left(-a, i, c \cdot z\right), \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 66.3

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 66.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around 0

       \[\leadsto \left(x \cdot y\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 61.5%

       \[\leadsto \left(x \cdot y\right) \cdot z \]

   if -1.8e212 < x < -1.60000000000000007e94 or 2.09999999999999979e31 < x

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.5%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(t \cdot x\right)} \]

   if -1.60000000000000007e94 < x < 2.59999999999999995e-20

   1. Initial program 78.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.1%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]

   if 2.59999999999999995e-20 < x < 2.09999999999999979e31

   1. Initial program 54.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \left(-i\right) \cdot \color{blue}{\left(y \cdot j\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+212}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -1.6e+94)
     t_1
     (if (<= x 2e-160)
       (* (fma (- c) z (* i a)) b)
       (if (<= x 2.25e+20) (* (fma (- b) z (* j t)) c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -1.6e+94) {
		tmp = t_1;
	} else if (x <= 2e-160) {
		tmp = fma(-c, z, (i * a)) * b;
	} else if (x <= 2.25e+20) {
		tmp = fma(-b, z, (j * t)) * c;
	} else {
		tmp = t_1;
	}
	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)
	tmp = 0.0
	if (x <= -1.6e+94)
		tmp = t_1;
	elseif (x <= 2e-160)
		tmp = Float64(fma(Float64(-c), z, Float64(i * a)) * b);
	elseif (x <= 2.25e+20)
		tmp = Float64(fma(Float64(-b), z, Float64(j * t)) * c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.6e+94], t$95$1, If[LessEqual[x, 2e-160], N[(N[((-c) * z + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 2.25e+20], N[(N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000007e94 or 2.25e20 < x

   1. Initial program 74.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 75.6

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if -1.60000000000000007e94 < x < 2e-160

   1. Initial program 78.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. neg-mul-1 N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right)} \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot c\right) \cdot z + \color{blue}{a \cdot i}\right) \cdot b \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, z, a \cdot i\right)} \cdot b \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right) \cdot b \]

      17. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, z, a \cdot i\right) \cdot b \]

      18. lower-*.f64 52.2

          \[\leadsto \mathsf{fma}\left(-c, z, \color{blue}{a \cdot i}\right) \cdot b \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, a \cdot i\right) \cdot b} \]

   if 2e-160 < x < 2.25e20

   1. Initial program 73.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \cdot c \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot z} + j \cdot t\right) \cdot c \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, z, j \cdot t\right)} \cdot c \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, z, j \cdot t\right) \cdot c \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, z, j \cdot t\right) \cdot c \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(b\right), z, \color{blue}{t \cdot j}\right) \cdot c \]

      11. lower-*.f64 58.1

          \[\leadsto \mathsf{fma}\left(-b, z, \color{blue}{t \cdot j}\right) \cdot c \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, z, t \cdot j\right) \cdot c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot a\right) \cdot b\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -1.05e-97)
     t_1
     (if (<= x -6.8e-224)
       (* (* (- c) z) b)
       (if (<= x 2.3e-18) (* (* i b) a) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -1.05e-97) {
		tmp = t_1;
	} else if (x <= -6.8e-224) {
		tmp = (-c * z) * b;
	} else if (x <= 2.3e-18) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	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)
	tmp = 0.0
	if (x <= -1.05e-97)
		tmp = t_1;
	elseif (x <= -6.8e-224)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (x <= 2.3e-18)
		tmp = Float64(Float64(i * b) * a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.05e-97], t$95$1, If[LessEqual[x, -6.8e-224], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 2.3e-18], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0500000000000001e-97 or 2.3000000000000001e-18 < x

   1. Initial program 74.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 62.3

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   if -1.0500000000000001e-97 < x < -6.79999999999999984e-224

   1. Initial program 70.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   4. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot t\right), j, \mathsf{fma}\left(-b, \mathsf{fma}\left(-a, i, c \cdot z\right), \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 23.5

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 23.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 35.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 35.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around inf

       \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 34.2%

       \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot \color{blue}{b} \]

   if -6.79999999999999984e-224 < x < 2.3000000000000001e-18

   1. Initial program 81.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.6%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-224}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+89}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.3e+89)
   (* (* z y) x)
   (if (<= z 8.4e+79) (* (* (- a) t) x) (* (* 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 <= -2.3e+89) {
		tmp = (z * y) * x;
	} else if (z <= 8.4e+79) {
		tmp = (-a * t) * x;
	} else {
		tmp = (y * x) * 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 <= (-2.3d+89)) then
        tmp = (z * y) * x
    else if (z <= 8.4d+79) then
        tmp = (-a * t) * x
    else
        tmp = (y * x) * 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 <= -2.3e+89) {
		tmp = (z * y) * x;
	} else if (z <= 8.4e+79) {
		tmp = (-a * t) * x;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.3e+89:
		tmp = (z * y) * x
	elif z <= 8.4e+79:
		tmp = (-a * t) * x
	else:
		tmp = (y * x) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.3e+89)
		tmp = Float64(Float64(z * y) * x);
	elseif (z <= 8.4e+79)
		tmp = Float64(Float64(Float64(-a) * t) * x);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -2.3e+89)
		tmp = (z * y) * x;
	elseif (z <= 8.4e+79)
		tmp = (-a * t) * x;
	else
		tmp = (y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.3e+89], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 8.4e+79], N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999999e89

   1. Initial program 70.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 46.7

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 38.7%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -2.2999999999999999e89 < z < 8.40000000000000032e79

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 39.9

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 34.3%

      \[\leadsto \left(\left(-a\right) \cdot t\right) \cdot x \]

   if 8.40000000000000032e79 < z

   1. Initial program 66.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot t\right), j, \mathsf{fma}\left(-b, \mathsf{fma}\left(-a, i, c \cdot z\right), \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 23.2

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 23.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 68.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 68.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around 0

       \[\leadsto \left(x \cdot y\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 41.3%

       \[\leadsto \left(x \cdot y\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+89}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 28.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+89}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.3e+89)
   (* (* z y) x)
   (if (<= z 1.05e+79) (* (* (- t) x) a) (* (* y x) z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.3e+89) {
		tmp = (z * y) * x;
	} else if (z <= 1.05e+79) {
		tmp = (-t * x) * a;
	} else {
		tmp = (y * x) * 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 <= (-2.3d+89)) then
        tmp = (z * y) * x
    else if (z <= 1.05d+79) then
        tmp = (-t * x) * a
    else
        tmp = (y * x) * 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 <= -2.3e+89) {
		tmp = (z * y) * x;
	} else if (z <= 1.05e+79) {
		tmp = (-t * x) * a;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.3e+89:
		tmp = (z * y) * x
	elif z <= 1.05e+79:
		tmp = (-t * x) * a
	else:
		tmp = (y * x) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.3e+89)
		tmp = Float64(Float64(z * y) * x);
	elseif (z <= 1.05e+79)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -2.3e+89)
		tmp = (z * y) * x;
	elseif (z <= 1.05e+79)
		tmp = (-t * x) * a;
	else
		tmp = (y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.3e+89], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.05e+79], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999999e89

   1. Initial program 70.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 46.7

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 38.7%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -2.2999999999999999e89 < z < 1.05000000000000004e79

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.1%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(t \cdot x\right)} \]

   if 1.05000000000000004e79 < z

   1. Initial program 66.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot t\right), j, \mathsf{fma}\left(-b, \mathsf{fma}\left(-a, i, c \cdot z\right), \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 23.2

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 23.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 68.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 68.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around 0

       \[\leadsto \left(x \cdot y\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 41.3%

       \[\leadsto \left(x \cdot y\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+89}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-13}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -4e+102)
   (* (* y x) z)
   (if (<= x 5.3e-13) (* (* i b) a) (* (* z y) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -4e+102) {
		tmp = (y * x) * z;
	} else if (x <= 5.3e-13) {
		tmp = (i * b) * a;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= (-4d+102)) then
        tmp = (y * x) * z
    else if (x <= 5.3d-13) then
        tmp = (i * b) * a
    else
        tmp = (z * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -4e+102) {
		tmp = (y * x) * z;
	} else if (x <= 5.3e-13) {
		tmp = (i * b) * a;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -4e+102:
		tmp = (y * x) * z
	elif x <= 5.3e-13:
		tmp = (i * b) * a
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -4e+102)
		tmp = Float64(Float64(y * x) * z);
	elseif (x <= 5.3e-13)
		tmp = Float64(Float64(i * b) * a);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= -4e+102)
		tmp = (y * x) * z;
	elseif (x <= 5.3e-13)
		tmp = (i * b) * a;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -4e+102], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 5.3e-13], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999991e102

   1. Initial program 79.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot t\right), j, \mathsf{fma}\left(-b, \mathsf{fma}\left(-a, i, c \cdot z\right), \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\right)}}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right) \cdot x} + b \cdot i\right) \cdot a \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, x, b \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, x, b \cdot i\right) \cdot a \]

      7. lower-*.f64 50.7

         \[\leadsto \mathsf{fma}\left(-t, x, \color{blue}{b \cdot i}\right) \cdot a \]

   7. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right)} \cdot z \]

      3. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)}\right) \cdot z \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      9. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      14. lower-*.f64 50.1

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{x \cdot y}\right) \cdot z \]

   10. Applied rewrites 50.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, x \cdot y\right) \cdot z} \]

   11. Taylor expanded in c around 0

       \[\leadsto \left(x \cdot y\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 45.7%

       \[\leadsto \left(x \cdot y\right) \cdot z \]

   if -3.99999999999999991e102 < x < 5.2999999999999996e-13

   1. Initial program 78.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.2%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]

   if 5.2999999999999996e-13 < x

   1. Initial program 67.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 36.9%

      \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-13}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 29.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -4 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-13}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= x -4e+102) t_1 (if (<= x 5.3e-13) (* (* i b) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (x <= -4e+102) {
		tmp = t_1;
	} else if (x <= 5.3e-13) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * x
    if (x <= (-4d+102)) then
        tmp = t_1
    else if (x <= 5.3d-13) then
        tmp = (i * b) * a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * y) * x;
	double tmp;
	if (x <= -4e+102) {
		tmp = t_1;
	} else if (x <= 5.3e-13) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if x <= -4e+102:
		tmp = t_1
	elif x <= 5.3e-13:
		tmp = (i * b) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (x <= -4e+102)
		tmp = t_1;
	elseif (x <= 5.3e-13)
		tmp = Float64(Float64(i * b) * a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (x <= -4e+102)
		tmp = t_1;
	elseif (x <= 5.3e-13)
		tmp = (i * b) * a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4e+102], t$95$1, If[LessEqual[x, 5.3e-13], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999991e102 or 5.2999999999999996e-13 < x

   1. Initial program 72.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{-1 \cdot \left(a \cdot t\right)} + y \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot t} + y \cdot z\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot a, t, y \cdot z\right)} \cdot x \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \cdot x \]

      11. lower-*.f64 70.7

          \[\leadsto \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in a around 0

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 38.6%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -3.99999999999999991e102 < x < 5.2999999999999996e-13

   1. Initial program 78.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.2%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-13}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 28.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-120}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+86}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -8.5e-120)
   (* (* b a) i)
   (if (<= b 7.6e+86) (* (* j t) c) (* (* i b) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -8.5e-120) {
		tmp = (b * a) * i;
	} else if (b <= 7.6e+86) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-8.5d-120)) then
        tmp = (b * a) * i
    else if (b <= 7.6d+86) then
        tmp = (j * t) * c
    else
        tmp = (i * b) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -8.5e-120) {
		tmp = (b * a) * i;
	} else if (b <= 7.6e+86) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -8.5e-120:
		tmp = (b * a) * i
	elif b <= 7.6e+86:
		tmp = (j * t) * c
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -8.5e-120)
		tmp = Float64(Float64(b * a) * i);
	elseif (b <= 7.6e+86)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -8.5e-120)
		tmp = (b * a) * i;
	elseif (b <= 7.6e+86)
		tmp = (j * t) * c;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -8.5e-120], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 7.6e+86], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.50000000000000059e-120

   1. Initial program 71.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.2%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 31.2%

       \[\leadsto \left(b \cdot a\right) \cdot i \]

   if -8.50000000000000059e-120 < b < 7.59999999999999956e86

   1. Initial program 76.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.3%

      \[\leadsto \left(t \cdot j\right) \cdot \color{blue}{c} \]

   if 7.59999999999999956e86 < b

   1. Initial program 82.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

      10. *-lft-identity N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

   5. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-120}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+86}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 22.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot a\right) \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* b a) i))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (b * a) * 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 = (b * a) * 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 (b * a) * i;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (b * a) * i

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(b * a) * i)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (b * a) * i;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

   3. cancel-sign-sub N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

   9. distribute-lft-in N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

   10. *-lft-identity N/A

       \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

   12. cancel-sign-sub-inv N/A

       \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   13. +-commutative N/A

       \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

5. Applied rewrites 58.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

8. Applied rewrites 22.5%

   \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
9. Step-by-step derivation

10. Applied rewrites 22.5%

    \[\leadsto \left(b \cdot a\right) \cdot i \]
11. Add Preprocessing

Alternative 28: 22.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot a\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i a) b))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * a) * b;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (i * a) * b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * a) * b;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * a) * b

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * a) * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * a) * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * a), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

   3. cancel-sign-sub N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + a \cdot \left(b \cdot i\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(j \cdot \left(c \cdot t - i \cdot y\right) + -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + \left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(b \cdot i\right)\right) \]

   9. distribute-lft-in N/A

      \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + b \cdot i\right)} \]

   10. *-lft-identity N/A

       \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(b \cdot i\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(b \cdot i\right)\right) \]

   12. cancel-sign-sub-inv N/A

       \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   13. +-commutative N/A

       \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

5. Applied rewrites 58.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-x, t, b \cdot i\right), a, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

8. Applied rewrites 22.5%

   \[\leadsto \left(b \cdot i\right) \cdot \color{blue}{a} \]
9. Step-by-step derivation

10. Applied rewrites 22.1%

    \[\leadsto \left(a \cdot i\right) \cdot b \]

11. Final simplification 22.1%

    \[\leadsto \left(i \cdot a\right) \cdot b \]
12. Add Preprocessing

Developer Target 1: 67.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))