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

Percentage Accurate: 73.3% → 82.8%

Time: 17.0s

Alternatives: 18

Speedup: 0.5×

• 57.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 0.5× speedup

Math

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

FPCore

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

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 * ((t * i) - (z * c)))) + (j * ((a * c) - (y * i)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = a * fma(j, c, (t * -x));
	}
	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(t * i) - Float64(z * c)))) + Float64(j * Float64(Float64(a * c) - Float64(y * i))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(a * fma(j, c, Float64(t * Float64(-x))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 52.8

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

   5. Applied rewrites 52.8%

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

4. Final simplification 86.7%

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

Alternative 2: 64.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - z \cdot c, y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -2.9e+106)
     t_1
     (if (<= t 4.3e+102)
       (fma b (- (* t i) (* z c)) (* y (fma j (- i) (* x z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -2.9e+106) {
		tmp = t_1;
	} else if (t <= 4.3e+102) {
		tmp = fma(b, ((t * i) - (z * c)), (y * fma(j, -i, (x * z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -2.9e+106)
		tmp = t_1;
	elseif (t <= 4.3e+102)
		tmp = fma(b, Float64(Float64(t * i) - Float64(z * c)), Float64(y * fma(j, Float64(-i), Float64(x * z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.9e+106], t$95$1, If[LessEqual[t, 4.3e+102], N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision] + N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9000000000000002e106 or 4.3000000000000001e102 < t

   1. Initial program 70.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -2.9000000000000002e106 < t < 4.3000000000000001e102

   1. Initial program 82.1%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 70.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+106}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - z \cdot c, y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(i, \frac{y}{-a}, c\right)\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -7.2e+78)
     t_1
     (if (<= t -2.3e-274)
       (* c (- (* a j) (* z b)))
       (if (<= t 1.25e-174)
         (* j (* a (fma i (/ y (- a)) c)))
         (if (<= t 4.3e+32) (* z (fma x y (* b (- c)))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -7.2e+78) {
		tmp = t_1;
	} else if (t <= -2.3e-274) {
		tmp = c * ((a * j) - (z * b));
	} else if (t <= 1.25e-174) {
		tmp = j * (a * fma(i, (y / -a), c));
	} else if (t <= 4.3e+32) {
		tmp = z * fma(x, y, (b * -c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -7.2e+78)
		tmp = t_1;
	elseif (t <= -2.3e-274)
		tmp = Float64(c * Float64(Float64(a * j) - Float64(z * b)));
	elseif (t <= 1.25e-174)
		tmp = Float64(j * Float64(a * fma(i, Float64(y / Float64(-a)), c)));
	elseif (t <= 4.3e+32)
		tmp = Float64(z * fma(x, y, Float64(b * Float64(-c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e+78], t$95$1, If[LessEqual[t, -2.3e-274], N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e-174], N[(j * N[(a * N[(i * N[(y / (-a)), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e+32], N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.20000000000000039e78 or 4.2999999999999997e32 < t

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -7.20000000000000039e78 < t < -2.29999999999999996e-274

   1. Initial program 83.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -2.29999999999999996e-274 < t < 1.2500000000000001e-174

   1. Initial program 87.2%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto j \cdot \left(a \cdot \left(\color{blue}{\left(0 - \frac{i \cdot y}{a}\right)} + c\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto j \cdot \left(a \cdot \color{blue}{\left(0 - \left(\frac{i \cdot y}{a} - c\right)\right)}\right) \]

      5. unsub-neg N/A

         \[\leadsto j \cdot \left(a \cdot \left(0 - \color{blue}{\left(\frac{i \cdot y}{a} + \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto j \cdot \left(a \cdot \left(0 - \left(\frac{i \cdot y}{a} + \color{blue}{-1 \cdot c}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto j \cdot \left(a \cdot \left(0 - \color{blue}{\left(-1 \cdot c + \frac{i \cdot y}{a}\right)}\right)\right) \]

      8. neg-sub0 N/A

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

      9. lower-*.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto j \cdot \left(a \cdot \color{blue}{\left(0 - \left(-1 \cdot c + \frac{i \cdot y}{a}\right)\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto j \cdot \left(a \cdot \left(0 - \color{blue}{\left(\frac{i \cdot y}{a} + -1 \cdot c\right)}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto j \cdot \left(a \cdot \left(0 - \left(\frac{i \cdot y}{a} + \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}\right)\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto j \cdot \left(a \cdot \left(0 - \color{blue}{\left(\frac{i \cdot y}{a} - c\right)}\right)\right) \]

      14. associate-+l- N/A

          \[\leadsto j \cdot \left(a \cdot \color{blue}{\left(\left(0 - \frac{i \cdot y}{a}\right) + c\right)}\right) \]

      15. neg-sub0 N/A

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

      16. associate-/l* N/A

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

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

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

      18. lower-fma.f64 N/A

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

   8. Applied rewrites 68.7%

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

   if 1.2500000000000001e-174 < t < 4.2999999999999997e32

   1. Initial program 79.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. lower-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 69.1

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

   11. Applied rewrites 69.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;j \cdot \left(a \cdot \mathsf{fma}\left(i, \frac{y}{-a}, c\right)\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -7.2e+78)
     t_1
     (if (<= t -2.3e-274)
       (* c (- (* a j) (* z b)))
       (if (<= t 1.25e-174)
         (* j (- (* a c) (* y i)))
         (if (<= t 4.3e+32) (* z (fma x y (* b (- c)))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -7.2e+78) {
		tmp = t_1;
	} else if (t <= -2.3e-274) {
		tmp = c * ((a * j) - (z * b));
	} else if (t <= 1.25e-174) {
		tmp = j * ((a * c) - (y * i));
	} else if (t <= 4.3e+32) {
		tmp = z * fma(x, y, (b * -c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -7.2e+78)
		tmp = t_1;
	elseif (t <= -2.3e-274)
		tmp = Float64(c * Float64(Float64(a * j) - Float64(z * b)));
	elseif (t <= 1.25e-174)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	elseif (t <= 4.3e+32)
		tmp = Float64(z * fma(x, y, Float64(b * Float64(-c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e+78], t$95$1, If[LessEqual[t, -2.3e-274], N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e-174], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e+32], N[(z * N[(x * y + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.20000000000000039e78 or 4.2999999999999997e32 < t

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -7.20000000000000039e78 < t < -2.29999999999999996e-274

   1. Initial program 83.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -2.29999999999999996e-274 < t < 1.2500000000000001e-174

   1. Initial program 87.2%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 1.2500000000000001e-174 < t < 4.2999999999999997e32

   1. Initial program 79.2%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. lower-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 69.1

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

   11. Applied rewrites 69.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, y, b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -7.2e+78)
     t_1
     (if (<= t -2.3e-274)
       (* c (- (* a j) (* z b)))
       (if (<= t 1.25e-174)
         (* j (- (* a c) (* y i)))
         (if (<= t 4.3e+32) (* z (fma c (- b) (* x y))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -7.2e+78) {
		tmp = t_1;
	} else if (t <= -2.3e-274) {
		tmp = c * ((a * j) - (z * b));
	} else if (t <= 1.25e-174) {
		tmp = j * ((a * c) - (y * i));
	} else if (t <= 4.3e+32) {
		tmp = z * fma(c, -b, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -7.2e+78)
		tmp = t_1;
	elseif (t <= -2.3e-274)
		tmp = Float64(c * Float64(Float64(a * j) - Float64(z * b)));
	elseif (t <= 1.25e-174)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	elseif (t <= 4.3e+32)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e+78], t$95$1, If[LessEqual[t, -2.3e-274], N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e-174], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e+32], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.20000000000000039e78 or 4.2999999999999997e32 < t

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -7.20000000000000039e78 < t < -2.29999999999999996e-274

   1. Initial program 83.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -2.29999999999999996e-274 < t < 1.2500000000000001e-174

   1. Initial program 87.2%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 1.2500000000000001e-174 < t < 4.2999999999999997e32

   1. Initial program 79.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-166}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+63}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(z, -c, t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -7.2e+78)
     t_1
     (if (<= t -2.3e-274)
       (* c (- (* a j) (* z b)))
       (if (<= t 1.25e-166)
         (* j (- (* a c) (* y i)))
         (if (<= t 9e+63) (* b (fma z (- c) (* t i))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -7.2e+78) {
		tmp = t_1;
	} else if (t <= -2.3e-274) {
		tmp = c * ((a * j) - (z * b));
	} else if (t <= 1.25e-166) {
		tmp = j * ((a * c) - (y * i));
	} else if (t <= 9e+63) {
		tmp = b * fma(z, -c, (t * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -7.2e+78)
		tmp = t_1;
	elseif (t <= -2.3e-274)
		tmp = Float64(c * Float64(Float64(a * j) - Float64(z * b)));
	elseif (t <= 1.25e-166)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	elseif (t <= 9e+63)
		tmp = Float64(b * fma(z, Float64(-c), Float64(t * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e+78], t$95$1, If[LessEqual[t, -2.3e-274], N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e-166], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e+63], N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.20000000000000039e78 or 9.00000000000000034e63 < t

   1. Initial program 72.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -7.20000000000000039e78 < t < -2.29999999999999996e-274

   1. Initial program 83.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -2.29999999999999996e-274 < t < 1.25e-166

   1. Initial program 87.6%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.25e-166 < t < 9.00000000000000034e63

   1. Initial program 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 77.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 55.0

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

   7. Applied rewrites 55.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-166}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+63}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(z, -c, t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(a \cdot j - z \cdot b\right)\\ t_2 := i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{if}\;i \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3.15 \cdot 10^{-168}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;i \leq 1.02 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (- (* a j) (* z b)))) (t_2 (* i (fma j (- y) (* t b)))))
   (if (<= i -2.7e-7)
     t_2
     (if (<= i -5e-270)
       t_1
       (if (<= i 3.15e-168)
         (* a (fma j c (* t (- x))))
         (if (<= i 1.02e+155) 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 * ((a * j) - (z * b));
	double t_2 = i * fma(j, -y, (t * b));
	double tmp;
	if (i <= -2.7e-7) {
		tmp = t_2;
	} else if (i <= -5e-270) {
		tmp = t_1;
	} else if (i <= 3.15e-168) {
		tmp = a * fma(j, c, (t * -x));
	} else if (i <= 1.02e+155) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(Float64(a * j) - Float64(z * b)))
	t_2 = Float64(i * fma(j, Float64(-y), Float64(t * b)))
	tmp = 0.0
	if (i <= -2.7e-7)
		tmp = t_2;
	elseif (i <= -5e-270)
		tmp = t_1;
	elseif (i <= 3.15e-168)
		tmp = Float64(a * fma(j, c, Float64(t * Float64(-x))));
	elseif (i <= 1.02e+155)
		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[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.7e-7], t$95$2, If[LessEqual[i, -5e-270], t$95$1, If[LessEqual[i, 3.15e-168], N[(a * N[(j * c + N[(t * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.02e+155], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.70000000000000009e-7 or 1.02e155 < i

   1. Initial program 75.2%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -2.70000000000000009e-7 < i < -4.9999999999999998e-270 or 3.14999999999999979e-168 < i < 1.02e155

   1. Initial program 79.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if -4.9999999999999998e-270 < i < 3.14999999999999979e-168

   1. Initial program 82.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 68.3

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

   5. Applied rewrites 68.3%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{elif}\;i \leq -5 \cdot 10^{-270}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{elif}\;i \leq 3.15 \cdot 10^{-168}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;i \leq 1.02 \cdot 10^{+155}:\\ \;\;\;\;c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;j \leq -1.95 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.45 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* a c) (* y i)))))
   (if (<= j -1.95e+134)
     t_1
     (if (<= j 1.45e+38) (fma b (fma z (- c) (* t i)) (* y (* x z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((a * c) - (y * i));
	double tmp;
	if (j <= -1.95e+134) {
		tmp = t_1;
	} else if (j <= 1.45e+38) {
		tmp = fma(b, fma(z, -c, (t * i)), (y * (x * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	tmp = 0.0
	if (j <= -1.95e+134)
		tmp = t_1;
	elseif (j <= 1.45e+38)
		tmp = fma(b, fma(z, Float64(-c), Float64(t * i)), Float64(y * Float64(x * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.95e+134], t$95$1, If[LessEqual[j, 1.45e+38], N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.94999999999999992e134 or 1.45000000000000003e38 < j

   1. Initial program 80.7%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -1.94999999999999992e134 < j < 1.45000000000000003e38

   1. Initial program 76.6%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in j around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.7

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

   8. Applied rewrites 61.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.95 \cdot 10^{+134}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 1.45 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), y \cdot \left(x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(b \cdot i\right)\\ t_2 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -44000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-276}:\\ \;\;\;\;a \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 10^{-223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+32}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (* b i))) (t_2 (* x (* y z))))
   (if (<= z -44000000000.0)
     t_2
     (if (<= z -8.2e-124)
       t_1
       (if (<= z -3e-276)
         (* a (* t (- x)))
         (if (<= z 1e-223) t_1 (if (<= z 7.2e+32) (* c (* a 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 = t * (b * i);
	double t_2 = x * (y * z);
	double tmp;
	if (z <= -44000000000.0) {
		tmp = t_2;
	} else if (z <= -8.2e-124) {
		tmp = t_1;
	} else if (z <= -3e-276) {
		tmp = a * (t * -x);
	} else if (z <= 1e-223) {
		tmp = t_1;
	} else if (z <= 7.2e+32) {
		tmp = c * (a * j);
	} 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 = t * (b * i)
    t_2 = x * (y * z)
    if (z <= (-44000000000.0d0)) then
        tmp = t_2
    else if (z <= (-8.2d-124)) then
        tmp = t_1
    else if (z <= (-3d-276)) then
        tmp = a * (t * -x)
    else if (z <= 1d-223) then
        tmp = t_1
    else if (z <= 7.2d+32) then
        tmp = c * (a * j)
    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 = t * (b * i);
	double t_2 = x * (y * z);
	double tmp;
	if (z <= -44000000000.0) {
		tmp = t_2;
	} else if (z <= -8.2e-124) {
		tmp = t_1;
	} else if (z <= -3e-276) {
		tmp = a * (t * -x);
	} else if (z <= 1e-223) {
		tmp = t_1;
	} else if (z <= 7.2e+32) {
		tmp = c * (a * j);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = t * (b * i)
	t_2 = x * (y * z)
	tmp = 0
	if z <= -44000000000.0:
		tmp = t_2
	elif z <= -8.2e-124:
		tmp = t_1
	elif z <= -3e-276:
		tmp = a * (t * -x)
	elif z <= 1e-223:
		tmp = t_1
	elif z <= 7.2e+32:
		tmp = c * (a * j)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * Float64(b * i))
	t_2 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (z <= -44000000000.0)
		tmp = t_2;
	elseif (z <= -8.2e-124)
		tmp = t_1;
	elseif (z <= -3e-276)
		tmp = Float64(a * Float64(t * Float64(-x)));
	elseif (z <= 1e-223)
		tmp = t_1;
	elseif (z <= 7.2e+32)
		tmp = Float64(c * Float64(a * j));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = t * (b * i);
	t_2 = x * (y * z);
	tmp = 0.0;
	if (z <= -44000000000.0)
		tmp = t_2;
	elseif (z <= -8.2e-124)
		tmp = t_1;
	elseif (z <= -3e-276)
		tmp = a * (t * -x);
	elseif (z <= 1e-223)
		tmp = t_1;
	elseif (z <= 7.2e+32)
		tmp = c * (a * j);
	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[(t * N[(b * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -44000000000.0], t$95$2, If[LessEqual[z, -8.2e-124], t$95$1, If[LessEqual[z, -3e-276], N[(a * N[(t * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-223], t$95$1, If[LessEqual[z, 7.2e+32], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.4e10 or 7.1999999999999994e32 < z

   1. Initial program 68.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 44.6

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

   8. Applied rewrites 44.6%

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

   if -4.4e10 < z < -8.2000000000000008e-124 or -2.99999999999999988e-276 < z < 9.9999999999999997e-224

   1. Initial program 84.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if -8.2000000000000008e-124 < z < -2.99999999999999988e-276

   1. Initial program 92.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in j around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 41.4

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

   8. Applied rewrites 41.4%

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

   if 9.9999999999999997e-224 < z < 7.1999999999999994e32

   1. Initial program 86.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.8

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

   8. Applied rewrites 31.8%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 42.1

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

   10. Applied rewrites 42.1%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -44000000000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-124}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-276}:\\ \;\;\;\;a \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 10^{-223}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+32}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+130}:\\ \;\;\;\;b \cdot \left(z \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= z -3.5e+130)
     (* b (* z (- c)))
     (if (<= z -1.45e+33)
       t_1
       (if (<= z 1.65e+103) (* a (fma j c (* t (- x)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (z <= -3.5e+130) {
		tmp = b * (z * -c);
	} else if (z <= -1.45e+33) {
		tmp = t_1;
	} else if (z <= 1.65e+103) {
		tmp = a * fma(j, c, (t * -x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (z <= -3.5e+130)
		tmp = Float64(b * Float64(z * Float64(-c)));
	elseif (z <= -1.45e+33)
		tmp = t_1;
	elseif (z <= 1.65e+103)
		tmp = Float64(a * fma(j, c, Float64(t * Float64(-x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e+130], N[(b * N[(z * (-c)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.45e+33], t$95$1, If[LessEqual[z, 1.65e+103], N[(a * N[(j * c + N[(t * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5000000000000001e130

   1. Initial program 64.4%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 51.0

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

   8. Applied rewrites 51.0%

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

      1. lift-neg.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 51.3

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

   10. Applied rewrites 51.3%

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

   if -3.5000000000000001e130 < z < -1.45000000000000012e33 or 1.65000000000000004e103 < z

   1. Initial program 73.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 56.3

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

   8. Applied rewrites 56.3%

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

   if -1.45000000000000012e33 < z < 1.65000000000000004e103

   1. Initial program 84.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 47.4

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

   5. Applied rewrites 47.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+130}:\\ \;\;\;\;b \cdot \left(z \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+56}:\\ \;\;\;\;a \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -8e+56)
   (* a (* t (- x)))
   (if (<= t 1.65e-153)
     (* x (* y z))
     (if (<= t 1.2e+15) (* z (* b (- c))) (* b (* t i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -8e+56) {
		tmp = a * (t * -x);
	} else if (t <= 1.65e-153) {
		tmp = x * (y * z);
	} else if (t <= 1.2e+15) {
		tmp = z * (b * -c);
	} else {
		tmp = b * (t * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (t <= (-8d+56)) then
        tmp = a * (t * -x)
    else if (t <= 1.65d-153) then
        tmp = x * (y * z)
    else if (t <= 1.2d+15) then
        tmp = z * (b * -c)
    else
        tmp = b * (t * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -8e+56) {
		tmp = a * (t * -x);
	} else if (t <= 1.65e-153) {
		tmp = x * (y * z);
	} else if (t <= 1.2e+15) {
		tmp = z * (b * -c);
	} else {
		tmp = b * (t * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -8e+56:
		tmp = a * (t * -x)
	elif t <= 1.65e-153:
		tmp = x * (y * z)
	elif t <= 1.2e+15:
		tmp = z * (b * -c)
	else:
		tmp = b * (t * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -8e+56)
		tmp = Float64(a * Float64(t * Float64(-x)));
	elseif (t <= 1.65e-153)
		tmp = Float64(x * Float64(y * z));
	elseif (t <= 1.2e+15)
		tmp = Float64(z * Float64(b * Float64(-c)));
	else
		tmp = Float64(b * Float64(t * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= -8e+56)
		tmp = a * (t * -x);
	elseif (t <= 1.65e-153)
		tmp = x * (y * z);
	elseif (t <= 1.2e+15)
		tmp = z * (b * -c);
	else
		tmp = b * (t * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -8e+56], N[(a * N[(t * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65e-153], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e+15], N[(z * N[(b * (-c)), $MachinePrecision]), $MachinePrecision], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.00000000000000074e56

   1. Initial program 76.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in j around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 54.1

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

   8. Applied rewrites 54.1%

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

   if -8.00000000000000074e56 < t < 1.64999999999999994e-153

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 38.3

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

   8. Applied rewrites 38.3%

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

   if 1.64999999999999994e-153 < t < 1.2e15

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 48.4

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

   8. Applied rewrites 48.4%

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

   if 1.2e15 < t

   1. Initial program 69.8%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 55.4

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 37.3

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

   11. Applied rewrites 37.3%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 40.7

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

   13. Applied rewrites 40.7%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+56}:\\ \;\;\;\;a \cdot \left(t \cdot \left(-x\right)\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(b \cdot \left(-c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-41}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* t i) (* z c)))))
   (if (<= b -4.3e-26)
     t_1
     (if (<= b 9.5e-41) (* a (fma j c (* t (- x)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((t * i) - (z * c));
	double tmp;
	if (b <= -4.3e-26) {
		tmp = t_1;
	} else if (b <= 9.5e-41) {
		tmp = a * fma(j, c, (t * -x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(t * i) - Float64(z * c)))
	tmp = 0.0
	if (b <= -4.3e-26)
		tmp = t_1;
	elseif (b <= 9.5e-41)
		tmp = Float64(a * fma(j, c, Float64(t * Float64(-x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.3e-26], t$95$1, If[LessEqual[b, 9.5e-41], N[(a * N[(j * c + N[(t * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.29999999999999988e-26 or 9.4999999999999997e-41 < b

   1. Initial program 81.8%

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

   3. Taylor expanded in b around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if -4.29999999999999988e-26 < b < 9.4999999999999997e-41

   1. Initial program 74.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 50.6

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

   5. Applied rewrites 50.6%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-26}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-41}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -44000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-223}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+32}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= z -44000000000.0)
     t_1
     (if (<= z 1e-223) (* t (* b i)) (if (<= z 7.2e+32) (* c (* a j)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (z <= -44000000000.0) {
		tmp = t_1;
	} else if (z <= 1e-223) {
		tmp = t * (b * i);
	} else if (z <= 7.2e+32) {
		tmp = c * (a * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (z <= (-44000000000.0d0)) then
        tmp = t_1
    else if (z <= 1d-223) then
        tmp = t * (b * i)
    else if (z <= 7.2d+32) then
        tmp = c * (a * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (z <= -44000000000.0) {
		tmp = t_1;
	} else if (z <= 1e-223) {
		tmp = t * (b * i);
	} else if (z <= 7.2e+32) {
		tmp = c * (a * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if z <= -44000000000.0:
		tmp = t_1
	elif z <= 1e-223:
		tmp = t * (b * i)
	elif z <= 7.2e+32:
		tmp = c * (a * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (z <= -44000000000.0)
		tmp = t_1;
	elseif (z <= 1e-223)
		tmp = Float64(t * Float64(b * i));
	elseif (z <= 7.2e+32)
		tmp = Float64(c * Float64(a * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (z <= -44000000000.0)
		tmp = t_1;
	elseif (z <= 1e-223)
		tmp = t * (b * i);
	elseif (z <= 7.2e+32)
		tmp = c * (a * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -44000000000.0], t$95$1, If[LessEqual[z, 1e-223], N[(t * N[(b * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e+32], N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4e10 or 7.1999999999999994e32 < z

   1. Initial program 68.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 44.6

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

   8. Applied rewrites 44.6%

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

   if -4.4e10 < z < 9.9999999999999997e-224

   1. Initial program 88.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.1

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

   8. Applied rewrites 39.1%

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

   if 9.9999999999999997e-224 < z < 7.1999999999999994e32

   1. Initial program 86.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

      2. lower-*.f64 31.8

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

   8. Applied rewrites 31.8%

      \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot j\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j\right) \cdot c} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot a\right)} \cdot c \]

      4. lower-*.f64 42.1

         \[\leadsto \color{blue}{\left(j \cdot a\right)} \cdot c \]

   10. Applied rewrites 42.1%

       \[\leadsto \color{blue}{\left(j \cdot a\right) \cdot c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -44000000000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 10^{-223}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+32}:\\ \;\;\;\;c \cdot \left(a \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -32000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= z -32000000000.0) t_1 (if (<= z 9.5e+45) (* i (* t b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (z <= -32000000000.0) {
		tmp = t_1;
	} else if (z <= 9.5e+45) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (z <= (-32000000000.0d0)) then
        tmp = t_1
    else if (z <= 9.5d+45) then
        tmp = i * (t * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (z <= -32000000000.0) {
		tmp = t_1;
	} else if (z <= 9.5e+45) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if z <= -32000000000.0:
		tmp = t_1
	elif z <= 9.5e+45:
		tmp = i * (t * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (z <= -32000000000.0)
		tmp = t_1;
	elseif (z <= 9.5e+45)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (z <= -32000000000.0)
		tmp = t_1;
	elseif (z <= 9.5e+45)
		tmp = i * (t * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -32000000000.0], t$95$1, If[LessEqual[z, 9.5e+45], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e10 or 9.4999999999999998e45 < z

   1. Initial program 68.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t\right)\right)} + y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot t\right)} + y \cdot z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot t, y \cdot z\right)} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      9. lower-*.f64 54.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, -t, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 46.3

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Applied rewrites 46.3%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if -3.2e10 < z < 9.4999999999999998e45

   1. Initial program 87.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. lower-*.f64 51.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot b\right)} \cdot t \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

      5. lower-*.f64 33.4

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Applied rewrites 33.4%

      \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -32000000000:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(b \cdot i\right)\\ \mathbf{if}\;b \leq -0.0034:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+59}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (* b i))))
   (if (<= b -0.0034) t_1 (if (<= b 9e+59) (* j (* a 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 = t * (b * i);
	double tmp;
	if (b <= -0.0034) {
		tmp = t_1;
	} else if (b <= 9e+59) {
		tmp = j * (a * c);
	} 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 * (b * i)
    if (b <= (-0.0034d0)) then
        tmp = t_1
    else if (b <= 9d+59) then
        tmp = j * (a * c)
    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 * (b * i);
	double tmp;
	if (b <= -0.0034) {
		tmp = t_1;
	} else if (b <= 9e+59) {
		tmp = j * (a * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = t * (b * i)
	tmp = 0
	if b <= -0.0034:
		tmp = t_1
	elif b <= 9e+59:
		tmp = j * (a * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * Float64(b * i))
	tmp = 0.0
	if (b <= -0.0034)
		tmp = t_1;
	elseif (b <= 9e+59)
		tmp = Float64(j * Float64(a * c));
	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 * (b * i);
	tmp = 0.0;
	if (b <= -0.0034)
		tmp = t_1;
	elseif (b <= 9e+59)
		tmp = j * (a * c);
	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[(t * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0034], t$95$1, If[LessEqual[b, 9e+59], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00339999999999999981 or 8.99999999999999919e59 < b

   1. Initial program 82.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto t \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \left(a \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \left(a \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto t \cdot \left(a \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto t \cdot \mathsf{fma}\left(a, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 53.0

          \[\leadsto t \cdot \mathsf{fma}\left(a, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(a, -x, i \cdot b\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto t \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(i \cdot b\right)} \]

      2. lower-*.f64 41.8

         \[\leadsto t \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Applied rewrites 41.8%

      \[\leadsto t \cdot \color{blue}{\left(i \cdot b\right)} \]

   if -0.00339999999999999981 < b < 8.99999999999999919e59

   1. Initial program 74.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. lower-neg.f64 46.7

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Applied rewrites 46.7%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot j\right) \cdot c} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot a\right)} \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c\right)} \]

      6. lower-*.f64 27.0

         \[\leadsto j \cdot \color{blue}{\left(a \cdot c\right)} \]

   8. Applied rewrites 27.0%

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0034:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+59}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{+45}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;a \leq 860000000000:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -1.26e+45)
   (* a (* c j))
   (if (<= a 860000000000.0) (* i (* t b)) (* j (* a c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -1.26e+45) {
		tmp = a * (c * j);
	} else if (a <= 860000000000.0) {
		tmp = i * (t * b);
	} else {
		tmp = j * (a * c);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-1.26d+45)) then
        tmp = a * (c * j)
    else if (a <= 860000000000.0d0) then
        tmp = i * (t * b)
    else
        tmp = j * (a * c)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -1.26e+45) {
		tmp = a * (c * j);
	} else if (a <= 860000000000.0) {
		tmp = i * (t * b);
	} else {
		tmp = j * (a * c);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -1.26e+45:
		tmp = a * (c * j)
	elif a <= 860000000000.0:
		tmp = i * (t * b)
	else:
		tmp = j * (a * c)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -1.26e+45)
		tmp = Float64(a * Float64(c * j));
	elseif (a <= 860000000000.0)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = Float64(j * Float64(a * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -1.26e+45)
		tmp = a * (c * j);
	elseif (a <= 860000000000.0)
		tmp = i * (t * b);
	else
		tmp = j * (a * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -1.26e+45], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 860000000000.0], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.26e45

   1. Initial program 67.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. lower-neg.f64 66.5

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

      2. lower-*.f64 36.3

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

   8. Applied rewrites 36.3%

      \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

   if -1.26e45 < a < 8.6e11

   1. Initial program 86.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. lower-*.f64 48.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot b\right)} \cdot t \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

      5. lower-*.f64 30.5

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Applied rewrites 30.5%

      \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

   if 8.6e11 < a

   1. Initial program 69.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. lower-neg.f64 56.8

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot j\right) \cdot c} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot a\right)} \cdot c \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c\right)} \]

      6. lower-*.f64 37.5

         \[\leadsto j \cdot \color{blue}{\left(a \cdot c\right)} \]

   8. Applied rewrites 37.5%

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{+45}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;a \leq 860000000000:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(a \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(t \cdot b\right)\\ \mathbf{if}\;b \leq -0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* t b))))
   (if (<= b -0.001) t_1 (if (<= b 1.05e+60) (* a (* c j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (t * b);
	double tmp;
	if (b <= -0.001) {
		tmp = t_1;
	} else if (b <= 1.05e+60) {
		tmp = a * (c * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = i * (t * b)
    if (b <= (-0.001d0)) then
        tmp = t_1
    else if (b <= 1.05d+60) then
        tmp = a * (c * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (t * b);
	double tmp;
	if (b <= -0.001) {
		tmp = t_1;
	} else if (b <= 1.05e+60) {
		tmp = a * (c * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = i * (t * b)
	tmp = 0
	if b <= -0.001:
		tmp = t_1
	elif b <= 1.05e+60:
		tmp = a * (c * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(t * b))
	tmp = 0.0
	if (b <= -0.001)
		tmp = t_1;
	elseif (b <= 1.05e+60)
		tmp = Float64(a * Float64(c * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = i * (t * b);
	tmp = 0.0;
	if (b <= -0.001)
		tmp = t_1;
	elseif (b <= 1.05e+60)
		tmp = a * (c * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.001], t$95$1, If[LessEqual[b, 1.05e+60], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1e-3 or 1.0500000000000001e60 < b

   1. Initial program 82.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. lower-*.f64 52.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Applied rewrites 52.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot b\right)} \cdot t \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

      5. lower-*.f64 41.0

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Applied rewrites 41.0%

      \[\leadsto \color{blue}{i \cdot \left(b \cdot t\right)} \]

   if -1e-3 < b < 1.0500000000000001e60

   1. Initial program 74.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. lower-neg.f64 46.7

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Applied rewrites 46.7%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

      2. lower-*.f64 26.9

         \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

   8. Applied rewrites 26.9%

      \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.001:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 23.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(c \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* c j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (c * j);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = a * (c * j)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (c * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (c * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(c * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (c * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

   2. +-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

   7. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

   10. lower-neg.f64 39.2

       \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

5. Applied rewrites 39.2%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

6. Taylor expanded in j around inf

   \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

   2. lower-*.f64 21.4

      \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

8. Applied rewrites 21.4%

   \[\leadsto a \cdot \color{blue}{\left(j \cdot c\right)} \]

9. Final simplification 21.4%

   \[\leadsto a \cdot \left(c \cdot j\right) \]
10. Add Preprocessing

Developer Target 1: 59.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024214 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))