Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.1% → 82.7%

Time: 19.8s

Alternatives: 27

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 53.3

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

   8. Applied rewrites 53.3%

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

4. Final simplification 83.9%

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

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma c (fma j t (* z (- b))) (* x (- (* y z) (* t a))))))
   (if (<= c -6.8e+143)
     t_1
     (if (<= c 5.8e+74)
       (fma
        t
        (fma j c (* x (- a)))
        (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a i)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(c, fma(j, t, (z * -b)), (x * ((y * z) - (t * a))));
	double tmp;
	if (c <= -6.8e+143) {
		tmp = t_1;
	} else if (c <= 5.8e+74) {
		tmp = fma(t, fma(j, c, (x * -a)), fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(Float64(y * z) - Float64(t * a))))
	tmp = 0.0
	if (c <= -6.8e+143)
		tmp = t_1;
	elseif (c <= 5.8e+74)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.8e+143], t$95$1, If[LessEqual[c, 5.8e+74], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -6.79999999999999964e143 or 5.8000000000000005e74 < c

   1. Initial program 64.3%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   if -6.79999999999999964e143 < c < 5.8000000000000005e74

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - t \cdot a\\ t_2 := \mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot t\_1\right)\\ \mathbf{if}\;c \leq -3.5 \cdot 10^{-25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_1, i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\right)\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* y z) (* t a)))
        (t_2 (fma c (fma j t (* z (- b))) (* x t_1))))
   (if (<= c -3.5e-25)
     t_2
     (if (<= c 2.15e-45)
       (fma x t_1 (* i (fma j (- y) (* a b))))
       (if (<= c 1.02e+41)
         (fma t (fma j c (* x (- a))) (* b (fma c (- z) (* a i))))
         t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (y * z) - (t * a);
	double t_2 = fma(c, fma(j, t, (z * -b)), (x * t_1));
	double tmp;
	if (c <= -3.5e-25) {
		tmp = t_2;
	} else if (c <= 2.15e-45) {
		tmp = fma(x, t_1, (i * fma(j, -y, (a * b))));
	} else if (c <= 1.02e+41) {
		tmp = fma(t, fma(j, c, (x * -a)), (b * fma(c, -z, (a * i))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(y * z) - Float64(t * a))
	t_2 = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * t_1))
	tmp = 0.0
	if (c <= -3.5e-25)
		tmp = t_2;
	elseif (c <= 2.15e-45)
		tmp = fma(x, t_1, Float64(i * fma(j, Float64(-y), Float64(a * b))));
	elseif (c <= 1.02e+41)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(b * fma(c, Float64(-z), Float64(a * i))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.5e-25], t$95$2, If[LessEqual[c, 2.15e-45], N[(x * t$95$1 + N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.02e+41], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.5000000000000002e-25 or 1.01999999999999992e41 < c

   1. Initial program 68.0%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 78.5%

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

   if -3.5000000000000002e-25 < c < 2.1499999999999999e-45

   1. Initial program 78.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 72.7%

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

   if 2.1499999999999999e-45 < c < 1.01999999999999992e41

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 82.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;c \leq 2.15 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\right)\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(-b\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-307}:\\ \;\;\;\;i \cdot \left(a \cdot b\right) + c \cdot \mathsf{fma}\left(j, t, t\_1\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), c \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (- b))))
   (if (<= y -2.2e-19)
     (* y (fma z x (* i (- j))))
     (if (<= y -2.05e-307)
       (+ (* i (* a b)) (* c (fma j t t_1)))
       (if (<= y 6.8e+176)
         (fma t (fma j c (* x (- a))) (* c t_1))
         (* y (fma j (- i) (* x z))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * -b;
	double tmp;
	if (y <= -2.2e-19) {
		tmp = y * fma(z, x, (i * -j));
	} else if (y <= -2.05e-307) {
		tmp = (i * (a * b)) + (c * fma(j, t, t_1));
	} else if (y <= 6.8e+176) {
		tmp = fma(t, fma(j, c, (x * -a)), (c * t_1));
	} else {
		tmp = y * fma(j, -i, (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(-b))
	tmp = 0.0
	if (y <= -2.2e-19)
		tmp = Float64(y * fma(z, x, Float64(i * Float64(-j))));
	elseif (y <= -2.05e-307)
		tmp = Float64(Float64(i * Float64(a * b)) + Float64(c * fma(j, t, t_1)));
	elseif (y <= 6.8e+176)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(c * t_1));
	else
		tmp = Float64(y * fma(j, Float64(-i), Float64(x * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * (-b)), $MachinePrecision]}, If[LessEqual[y, -2.2e-19], N[(y * N[(z * x + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.05e-307], N[(N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(j * t + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+176], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(c * t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.1999999999999998e-19

   1. Initial program 64.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 65.7

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

   8. Applied rewrites 65.7%

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

   if -2.1999999999999998e-19 < y < -2.05000000000000016e-307

   1. Initial program 80.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if -2.05000000000000016e-307 < y < 6.80000000000000028e176

   1. Initial program 74.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 64.9

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

   8. Applied rewrites 64.9%

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

   if 6.80000000000000028e176 < y

   1. Initial program 77.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-307}:\\ \;\;\;\;i \cdot \left(a \cdot b\right) + c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), c \cdot \left(z \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(a, -x, c \cdot j\right)\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma z (fma b (- c) (* x y)) (* t (fma a (- x) (* c j))))))
   (if (<= t -1.8e-32)
     t_1
     (if (<= t 5.6e+38)
       (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a i))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(z, fma(b, -c, (x * y)), (t * fma(a, -x, (c * j))));
	double tmp;
	if (t <= -1.8e-32) {
		tmp = t_1;
	} else if (t <= 5.6e+38) {
		tmp = fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(z, fma(b, Float64(-c), Float64(x * y)), Float64(t * fma(a, Float64(-x), Float64(c * j))))
	tmp = 0.0
	if (t <= -1.8e-32)
		tmp = t_1;
	elseif (t <= 5.6e+38)
		tmp = fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t * N[(a * (-x) + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8e-32], t$95$1, If[LessEqual[t, 5.6e+38], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.79999999999999996e-32 or 5.6e38 < t

   1. Initial program 66.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 77.6%

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

   if -1.79999999999999996e-32 < t < 5.6e38

   1. Initial program 80.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

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

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. sub-neg N/A

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

   5. Applied rewrites 74.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(a, -x, c \cdot j\right)\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(a, -x, c \cdot j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;j \leq 5.8 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.5e-25)
   (+ (* j (- (* t c) (* y i))) (* i (* a b)))
   (if (<= j 5.8e+180)
     (fma c (fma j t (* z (- b))) (* x (- (* y z) (* t a))))
     (fma t (fma j c (* x (- a))) (* b (fma c (- z) (* a 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 (j <= -2.5e-25) {
		tmp = (j * ((t * c) - (y * i))) + (i * (a * b));
	} else if (j <= 5.8e+180) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * ((y * z) - (t * a))));
	} else {
		tmp = fma(t, fma(j, c, (x * -a)), (b * fma(c, -z, (a * i))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.5e-25)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(i * Float64(a * b)));
	elseif (j <= 5.8e+180)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(b * fma(c, Float64(-z), Float64(a * i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.5e-25], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 5.8e+180], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.49999999999999981e-25

   1. Initial program 74.3%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -2.49999999999999981e-25 < j < 5.80000000000000015e180

   1. Initial program 72.7%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 73.1%

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

   if 5.80000000000000015e180 < j

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

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

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

      18. mul-1-neg N/A

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

      19. lower-*.f64 N/A

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

   5. Applied rewrites 76.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;j \leq 5.8 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), a \cdot \left(b \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.5e-25)
   (+ (* j (- (* t c) (* y i))) (* i (* a b)))
   (if (<= j 2.9e+159)
     (fma c (fma j t (* z (- b))) (* x (- (* y z) (* t a))))
     (fma t (fma j c (* x (- a))) (* a (* b i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -2.5e-25) {
		tmp = (j * ((t * c) - (y * i))) + (i * (a * b));
	} else if (j <= 2.9e+159) {
		tmp = fma(c, fma(j, t, (z * -b)), (x * ((y * z) - (t * a))));
	} else {
		tmp = fma(t, fma(j, c, (x * -a)), (a * (b * i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.5e-25)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(i * Float64(a * b)));
	elseif (j <= 2.9e+159)
		tmp = fma(c, fma(j, t, Float64(z * Float64(-b))), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(a * Float64(b * i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.5e-25], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 2.9e+159], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.49999999999999981e-25

   1. Initial program 74.3%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -2.49999999999999981e-25 < j < 2.90000000000000014e159

   1. Initial program 72.5%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 73.5%

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

   if 2.90000000000000014e159 < j

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 72.2

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

   8. Applied rewrites 72.2%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;j \leq 2.9 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), a \cdot \left(b \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), a \cdot \left(b \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma y x (* b (- c))))))
   (if (<= z -1.1e+33)
     t_1
     (if (<= z -1.9e-38)
       (* y (fma j (- i) (* x z)))
       (if (<= z 9.5e+91) (fma t (fma j c (* x (- a))) (* a (* b i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(y, x, (b * -c));
	double tmp;
	if (z <= -1.1e+33) {
		tmp = t_1;
	} else if (z <= -1.9e-38) {
		tmp = y * fma(j, -i, (x * z));
	} else if (z <= 9.5e+91) {
		tmp = fma(t, fma(j, c, (x * -a)), (a * (b * i)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(y, x, Float64(b * Float64(-c))))
	tmp = 0.0
	if (z <= -1.1e+33)
		tmp = t_1;
	elseif (z <= -1.9e-38)
		tmp = Float64(y * fma(j, Float64(-i), Float64(x * z)));
	elseif (z <= 9.5e+91)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(a * Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(y * x + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+33], t$95$1, If[LessEqual[z, -1.9e-38], N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+91], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.09999999999999997e33 or 9.5000000000000001e91 < z

   1. Initial program 59.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if -1.09999999999999997e33 < z < -1.9e-38

   1. Initial program 86.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if -1.9e-38 < z < 9.5000000000000001e91

   1. Initial program 81.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, -i, x \cdot z\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), a \cdot \left(b \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma y x (* b (- c))))))
   (if (<= z -2.45e+61)
     t_1
     (if (<= z 1.7e+45) (+ (* j (- (* t c) (* y i))) (* i (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(y, x, (b * -c));
	double tmp;
	if (z <= -2.45e+61) {
		tmp = t_1;
	} else if (z <= 1.7e+45) {
		tmp = (j * ((t * c) - (y * i))) + (i * (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(y, x, Float64(b * Float64(-c))))
	tmp = 0.0
	if (z <= -2.45e+61)
		tmp = t_1;
	elseif (z <= 1.7e+45)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(i * Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(y * x + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.45e+61], t$95$1, If[LessEqual[z, 1.7e+45], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.45000000000000013e61 or 1.7e45 < z

   1. Initial program 62.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if -2.45000000000000013e61 < z < 1.7e45

   1. Initial program 80.8%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+61}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-269}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-16}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma t (- x) (* b i)))))
   (if (<= a -2.35e+42)
     t_1
     (if (<= a -1.55e-269)
       (* c (* z (- b)))
       (if (<= a 8.5e-217)
         (* y (* x z))
         (if (<= a 9e-16) (* c (* t j)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(t, -x, (b * i));
	double tmp;
	if (a <= -2.35e+42) {
		tmp = t_1;
	} else if (a <= -1.55e-269) {
		tmp = c * (z * -b);
	} else if (a <= 8.5e-217) {
		tmp = y * (x * z);
	} else if (a <= 9e-16) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(t, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (a <= -2.35e+42)
		tmp = t_1;
	elseif (a <= -1.55e-269)
		tmp = Float64(c * Float64(z * Float64(-b)));
	elseif (a <= 8.5e-217)
		tmp = Float64(y * Float64(x * z));
	elseif (a <= 9e-16)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.35e+42], t$95$1, If[LessEqual[a, -1.55e-269], N[(c * N[(z * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.5e-217], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-16], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.34999999999999993e42 or 9.0000000000000003e-16 < a

   1. Initial program 71.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if -2.34999999999999993e42 < a < -1.54999999999999983e-269

   1. Initial program 69.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} \]
   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. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 41.1

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

   8. Applied rewrites 41.1%

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

   if -1.54999999999999983e-269 < a < 8.4999999999999994e-217

   1. Initial program 80.8%

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

   3. Taylor expanded in z around 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 50.2

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

   5. Applied rewrites 50.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 43.7

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

   10. Applied rewrites 43.7%

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

   if 8.4999999999999994e-217 < a < 9.0000000000000003e-16

   1. Initial program 76.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in j around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 39.4

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

   11. Applied rewrites 39.4%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{+42}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-269}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-16}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.14 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \left(i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;i \leq -3.3 \cdot 10^{-180}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{+86}:\\ \;\;\;\;-a \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.14e+64)
   (* y (* i (- j)))
   (if (<= i -3.3e-180)
     (* j (* t c))
     (if (<= i 3.5e-166)
       (* y (* x z))
       (if (<= i 2.1e+86) (- (* a (* x t))) (* a (* b i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.14e+64) {
		tmp = y * (i * -j);
	} else if (i <= -3.3e-180) {
		tmp = j * (t * c);
	} else if (i <= 3.5e-166) {
		tmp = y * (x * z);
	} else if (i <= 2.1e+86) {
		tmp = -(a * (x * t));
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (i <= (-1.14d+64)) then
        tmp = y * (i * -j)
    else if (i <= (-3.3d-180)) then
        tmp = j * (t * c)
    else if (i <= 3.5d-166) then
        tmp = y * (x * z)
    else if (i <= 2.1d+86) then
        tmp = -(a * (x * t))
    else
        tmp = a * (b * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.14e+64) {
		tmp = y * (i * -j);
	} else if (i <= -3.3e-180) {
		tmp = j * (t * c);
	} else if (i <= 3.5e-166) {
		tmp = y * (x * z);
	} else if (i <= 2.1e+86) {
		tmp = -(a * (x * t));
	} else {
		tmp = a * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -1.14e+64:
		tmp = y * (i * -j)
	elif i <= -3.3e-180:
		tmp = j * (t * c)
	elif i <= 3.5e-166:
		tmp = y * (x * z)
	elif i <= 2.1e+86:
		tmp = -(a * (x * t))
	else:
		tmp = a * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.14e+64)
		tmp = Float64(y * Float64(i * Float64(-j)));
	elseif (i <= -3.3e-180)
		tmp = Float64(j * Float64(t * c));
	elseif (i <= 3.5e-166)
		tmp = Float64(y * Float64(x * z));
	elseif (i <= 2.1e+86)
		tmp = Float64(-Float64(a * Float64(x * t)));
	else
		tmp = Float64(a * Float64(b * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (i <= -1.14e+64)
		tmp = y * (i * -j);
	elseif (i <= -3.3e-180)
		tmp = j * (t * c);
	elseif (i <= 3.5e-166)
		tmp = y * (x * z);
	elseif (i <= 2.1e+86)
		tmp = -(a * (x * t));
	else
		tmp = a * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.14e+64], N[(y * N[(i * (-j)), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -3.3e-180], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.5e-166], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.1e+86], (-N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision]), N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -1.14e64

   1. Initial program 64.8%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 39.1

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

   8. Applied rewrites 39.1%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 40.9

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

   10. Applied rewrites 40.9%

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

   if -1.14e64 < i < -3.29999999999999998e-180

   1. Initial program 87.8%

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

   3. Taylor expanded in j around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.0

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

   8. Applied rewrites 43.0%

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

   if -3.29999999999999998e-180 < i < 3.4999999999999999e-166

   1. Initial program 76.4%

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

   3. Taylor expanded in 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 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 57.5

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

   7. Applied rewrites 57.5%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   if 3.4999999999999999e-166 < i < 2.0999999999999999e86

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 41.5

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 39.4

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

   8. Applied rewrites 39.4%

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

   if 2.0999999999999999e86 < i

   1. Initial program 57.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.6

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

   8. Applied rewrites 40.6%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.14 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \left(i \cdot \left(-j\right)\right)\\ \mathbf{elif}\;i \leq -3.3 \cdot 10^{-180}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{+86}:\\ \;\;\;\;-a \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -0.00018:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+93}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* z (- b)))))
   (if (<= z -2.1e+189)
     t_1
     (if (<= z -0.00018)
       (* z (* x y))
       (if (<= z 3.6e+40)
         (* c (* t j))
         (if (<= z 3e+93) (* t (* x (- a))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (z * -b);
	double tmp;
	if (z <= -2.1e+189) {
		tmp = t_1;
	} else if (z <= -0.00018) {
		tmp = z * (x * y);
	} else if (z <= 3.6e+40) {
		tmp = c * (t * j);
	} else if (z <= 3e+93) {
		tmp = t * (x * -a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (z * -b)
    if (z <= (-2.1d+189)) then
        tmp = t_1
    else if (z <= (-0.00018d0)) then
        tmp = z * (x * y)
    else if (z <= 3.6d+40) then
        tmp = c * (t * j)
    else if (z <= 3d+93) then
        tmp = t * (x * -a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (z * -b);
	double tmp;
	if (z <= -2.1e+189) {
		tmp = t_1;
	} else if (z <= -0.00018) {
		tmp = z * (x * y);
	} else if (z <= 3.6e+40) {
		tmp = c * (t * j);
	} else if (z <= 3e+93) {
		tmp = t * (x * -a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (z * -b)
	tmp = 0
	if z <= -2.1e+189:
		tmp = t_1
	elif z <= -0.00018:
		tmp = z * (x * y)
	elif z <= 3.6e+40:
		tmp = c * (t * j)
	elif z <= 3e+93:
		tmp = t * (x * -a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(z * Float64(-b)))
	tmp = 0.0
	if (z <= -2.1e+189)
		tmp = t_1;
	elseif (z <= -0.00018)
		tmp = Float64(z * Float64(x * y));
	elseif (z <= 3.6e+40)
		tmp = Float64(c * Float64(t * j));
	elseif (z <= 3e+93)
		tmp = Float64(t * Float64(x * Float64(-a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (z * -b);
	tmp = 0.0;
	if (z <= -2.1e+189)
		tmp = t_1;
	elseif (z <= -0.00018)
		tmp = z * (x * y);
	elseif (z <= 3.6e+40)
		tmp = c * (t * j);
	elseif (z <= 3e+93)
		tmp = t * (x * -a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(z * (-b)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+189], t$95$1, If[LessEqual[z, -0.00018], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e+40], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3e+93], N[(t * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.09999999999999992e189 or 2.99999999999999978e93 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around 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 73.8

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

   5. Applied rewrites 73.8%

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

   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. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 54.0

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

   8. Applied rewrites 54.0%

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

   if -2.09999999999999992e189 < z < -1.80000000000000011e-4

   1. Initial program 59.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.4

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

   8. Applied rewrites 41.4%

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

   if -1.80000000000000011e-4 < z < 3.59999999999999996e40

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in c around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 53.0

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

   8. Applied rewrites 53.0%

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

   9. Taylor expanded in j around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 35.3

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

   11. Applied rewrites 35.3%

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

   if 3.59999999999999996e40 < z < 2.99999999999999978e93

   1. Initial program 67.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 59.8

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

   8. Applied rewrites 59.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+189}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;z \leq -0.00018:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+93}:\\ \;\;\;\;t \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-240}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma y x (* b (- c))))))
   (if (<= z -2.2e+21)
     t_1
     (if (<= z -3.2e-240)
       (* i (fma j (- y) (* a b)))
       (if (<= z 9.5e+91) (* t (fma j c (* x (- a)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(y, x, (b * -c));
	double tmp;
	if (z <= -2.2e+21) {
		tmp = t_1;
	} else if (z <= -3.2e-240) {
		tmp = i * fma(j, -y, (a * b));
	} else if (z <= 9.5e+91) {
		tmp = t * fma(j, c, (x * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(y, x, Float64(b * Float64(-c))))
	tmp = 0.0
	if (z <= -2.2e+21)
		tmp = t_1;
	elseif (z <= -3.2e-240)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (z <= 9.5e+91)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(y * x + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+21], t$95$1, If[LessEqual[z, -3.2e-240], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+91], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e21 or 9.5000000000000001e91 < z

   1. Initial program 60.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if -2.2e21 < z < -3.1999999999999999e-240

   1. Initial program 80.7%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if -3.1999999999999999e-240 < z < 9.5000000000000001e91

   1. Initial program 83.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 57.8

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

   5. Applied rewrites 57.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-240}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, b \cdot \left(-c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-240}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (fma c (- b) (* x y)))))
   (if (<= z -5.5e-102)
     t_1
     (if (<= z -3.2e-240)
       (* i (fma j (- y) (* a b)))
       (if (<= z 9.5e+91) (* t (fma j c (* x (- a)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * fma(c, -b, (x * y));
	double tmp;
	if (z <= -5.5e-102) {
		tmp = t_1;
	} else if (z <= -3.2e-240) {
		tmp = i * fma(j, -y, (a * b));
	} else if (z <= 9.5e+91) {
		tmp = t * fma(j, c, (x * -a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * fma(c, Float64(-b), Float64(x * y)))
	tmp = 0.0
	if (z <= -5.5e-102)
		tmp = t_1;
	elseif (z <= -3.2e-240)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (z <= 9.5e+91)
		tmp = Float64(t * fma(j, c, Float64(x * Float64(-a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e-102], t$95$1, If[LessEqual[z, -3.2e-240], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+91], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4999999999999997e-102 or 9.5000000000000001e91 < z

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if -5.4999999999999997e-102 < z < -3.1999999999999999e-240

   1. Initial program 75.5%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if -3.1999999999999999e-240 < z < 9.5000000000000001e91

   1. Initial program 83.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 57.8

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

   5. Applied rewrites 57.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-102}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-240}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-156}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3e-25)
   (* c (fma j t (* z (- b))))
   (if (<= c -3.8e-156)
     (* i (fma j (- y) (* a b)))
     (if (<= c 2.3e+76)
       (* y (fma z x (* i (- j))))
       (* c (fma b (- z) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -3e-25) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= -3.8e-156) {
		tmp = i * fma(j, -y, (a * b));
	} else if (c <= 2.3e+76) {
		tmp = y * fma(z, x, (i * -j));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3e-25)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= -3.8e-156)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (c <= 2.3e+76)
		tmp = Float64(y * fma(z, x, Float64(i * Float64(-j))));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3e-25], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.8e-156], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.3e+76], N[(y * N[(z * x + N[(i * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.9999999999999998e-25

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -2.9999999999999998e-25 < c < -3.80000000000000008e-156

   1. Initial program 84.3%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   if -3.80000000000000008e-156 < c < 2.30000000000000001e76

   1. Initial program 76.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(z, x, \color{blue}{\mathsf{neg}\left(i \cdot j\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto y \cdot \mathsf{fma}\left(z, x, \color{blue}{i \cdot \left(\mathsf{neg}\left(j\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(z, x, i \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(z, x, \color{blue}{i \cdot \left(-1 \cdot j\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(z, x, i \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      10. lower-neg.f64 52.2

          \[\leadsto y \cdot \mathsf{fma}\left(z, x, i \cdot \color{blue}{\left(-j\right)}\right) \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)} \]

   if 2.30000000000000001e76 < c

   1. Initial program 58.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      6. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      8. lower-*.f64 69.4

         \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-156}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+76}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, x, i \cdot \left(-j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-237}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3e-25)
   (* c (fma j t (* z (- b))))
   (if (<= c -3.8e-237)
     (* i (fma j (- y) (* a b)))
     (if (<= c 1.16e+133)
       (* x (- (* y z) (* t a)))
       (* c (fma b (- z) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -3e-25) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= -3.8e-237) {
		tmp = i * fma(j, -y, (a * b));
	} else if (c <= 1.16e+133) {
		tmp = x * ((y * z) - (t * a));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3e-25)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= -3.8e-237)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (c <= 1.16e+133)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3e-25], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.8e-237], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.16e+133], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.9999999999999998e-25

   1. Initial program 73.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 70.4

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -2.9999999999999998e-25 < c < -3.80000000000000024e-237

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 54.4

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -3.80000000000000024e-237 < c < 1.1599999999999999e133

   1. Initial program 76.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. lower-*.f64 50.9

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Applied rewrites 50.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if 1.1599999999999999e133 < c

   1. Initial program 55.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      6. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      8. lower-*.f64 71.8

         \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-237}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-237}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3e-25)
   (* c (fma j t (* z (- b))))
   (if (<= c -7.5e-237)
     (* i (fma j (- y) (* a b)))
     (if (<= c 1.8e-12)
       (* a (fma t (- x) (* b i)))
       (* c (fma b (- z) (* t j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -3e-25) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= -7.5e-237) {
		tmp = i * fma(j, -y, (a * b));
	} else if (c <= 1.8e-12) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3e-25)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= -7.5e-237)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * b)));
	elseif (c <= 1.8e-12)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3e-25], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -7.5e-237], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.8e-12], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.9999999999999998e-25

   1. Initial program 73.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 70.4

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -2.9999999999999998e-25 < c < -7.50000000000000034e-237

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \mathsf{neg}\left(y\right), \color{blue}{b \cdot a}\right) \]

      12. lower-*.f64 54.4

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot a}\right) \]

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot a\right)} \]

   if -7.50000000000000034e-237 < c < 1.8e-12

   1. Initial program 77.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 45.9

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 45.9%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if 1.8e-12 < c

   1. Initial program 63.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      6. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      8. lower-*.f64 60.8

         \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   8. Applied rewrites 60.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-237}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -0.00018:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* z (- b)))))
   (if (<= z -2.1e+189)
     t_1
     (if (<= z -0.00018)
       (* z (* x y))
       (if (<= z 6.5e+69) (* c (* t j)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (z * -b);
	double tmp;
	if (z <= -2.1e+189) {
		tmp = t_1;
	} else if (z <= -0.00018) {
		tmp = z * (x * y);
	} else if (z <= 6.5e+69) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (z * -b)
    if (z <= (-2.1d+189)) then
        tmp = t_1
    else if (z <= (-0.00018d0)) then
        tmp = z * (x * y)
    else if (z <= 6.5d+69) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (z * -b);
	double tmp;
	if (z <= -2.1e+189) {
		tmp = t_1;
	} else if (z <= -0.00018) {
		tmp = z * (x * y);
	} else if (z <= 6.5e+69) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (z * -b)
	tmp = 0
	if z <= -2.1e+189:
		tmp = t_1
	elif z <= -0.00018:
		tmp = z * (x * y)
	elif z <= 6.5e+69:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(z * Float64(-b)))
	tmp = 0.0
	if (z <= -2.1e+189)
		tmp = t_1;
	elseif (z <= -0.00018)
		tmp = Float64(z * Float64(x * y));
	elseif (z <= 6.5e+69)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (z * -b);
	tmp = 0.0;
	if (z <= -2.1e+189)
		tmp = t_1;
	elseif (z <= -0.00018)
		tmp = z * (x * y);
	elseif (z <= 6.5e+69)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(z * (-b)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+189], t$95$1, If[LessEqual[z, -0.00018], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+69], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.09999999999999992e189 or 6.5000000000000001e69 < z

   1. Initial program 63.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 71.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot c\right) \cdot z}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(c \cdot b\right)} \cdot z\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{c \cdot \left(\mathsf{neg}\left(b \cdot z\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)} \]

      13. mul-1-neg N/A

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      14. lower-neg.f64 51.7

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right) \]

   8. Applied rewrites 51.7%

      \[\leadsto \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)} \]

   if -2.09999999999999992e189 < z < -1.80000000000000011e-4

   1. Initial program 59.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 59.7

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. lower-*.f64 41.4

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Applied rewrites 41.4%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -1.80000000000000011e-4 < z < 6.5000000000000001e69

   1. Initial program 82.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(c \cdot b\right)} \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \]

      14. lower-neg.f64 53.4

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right)\right) \]

   8. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)}\right) \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. lower-*.f64 34.8

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   11. Applied rewrites 34.8%

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+189}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;z \leq -0.00018:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(z \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.1e-95)
   (* c (fma j t (* z (- b))))
   (if (<= c 1.8e-12)
     (* a (fma t (- x) (* b i)))
     (* c (fma b (- z) (* t j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.1e-95) {
		tmp = c * fma(j, t, (z * -b));
	} else if (c <= 1.8e-12) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * fma(b, -z, (t * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.1e-95)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-b))));
	elseif (c <= 1.8e-12)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * fma(b, Float64(-z), Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.1e-95], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.8e-12], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.0999999999999999e-95

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(j, t, -1 \cdot \left(b \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, \color{blue}{b \cdot \left(-1 \cdot z\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

      10. lower-neg.f64 61.6

          \[\leadsto c \cdot \mathsf{fma}\left(j, t, b \cdot \color{blue}{\left(-z\right)}\right) \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(j, t, b \cdot \left(-z\right)\right)} \]

   if -1.0999999999999999e-95 < c < 1.8e-12

   1. Initial program 80.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 44.9

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 44.9%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   if 1.8e-12 < c

   1. Initial program 63.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      6. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      8. lower-*.f64 60.8

         \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   8. Applied rewrites 60.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-95}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \mathbf{if}\;c \leq -5800000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma b (- z) (* t j)))))
   (if (<= c -5800000000.0)
     t_1
     (if (<= c 1.8e-12) (* a (fma t (- x) (* b i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * fma(b, -z, (t * j));
	double tmp;
	if (c <= -5800000000.0) {
		tmp = t_1;
	} else if (c <= 1.8e-12) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(b, Float64(-z), Float64(t * j)))
	tmp = 0.0
	if (c <= -5800000000.0)
		tmp = t_1;
	elseif (c <= 1.8e-12)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(b * (-z) + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5800000000.0], t$95$1, If[LessEqual[c, 1.8e-12], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.8e9 or 1.8e-12 < c

   1. Initial program 67.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      2. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      4. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      6. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      8. lower-*.f64 65.8

         \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{j \cdot t}\right) \]

   8. Applied rewrites 65.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, j \cdot t\right)} \]

   if -5.8e9 < c < 1.8e-12

   1. Initial program 78.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 42.6

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 42.6%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5800000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+177}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* z (* x y))))
   (if (<= y -4.8e+122)
     t_1
     (if (<= y 1.55e+177) (* b (fma c (- z) (* a i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = z * (x * y);
	double tmp;
	if (y <= -4.8e+122) {
		tmp = t_1;
	} else if (y <= 1.55e+177) {
		tmp = b * fma(c, -z, (a * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(z * Float64(x * y))
	tmp = 0.0
	if (y <= -4.8e+122)
		tmp = t_1;
	elseif (y <= 1.55e+177)
		tmp = Float64(b * fma(c, Float64(-z), Float64(a * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+122], t$95$1, If[LessEqual[y, 1.55e+177], N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000004e122 or 1.55e177 < y

   1. Initial program 63.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 65.0

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. lower-*.f64 60.2

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -4.8000000000000004e122 < y < 1.55e177

   1. Initial program 76.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \]

      3. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \]

      9. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto b \cdot \left(c \cdot \left(-1 \cdot z\right) + \color{blue}{a \cdot i}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, a \cdot i\right)} \]

      15. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      16. lower-neg.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{i \cdot a}\right) \]

      18. lower-*.f64 42.5

          \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{i \cdot a}\right) \]

   5. Applied rewrites 42.5%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+122}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+177}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.12 \cdot 10^{+73}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2.45e+17) t_1 (if (<= j 1.12e+73) (* z (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.45e+17) {
		tmp = t_1;
	} else if (j <= 1.12e+73) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2.45d+17)) then
        tmp = t_1
    else if (j <= 1.12d+73) then
        tmp = z * (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.45e+17) {
		tmp = t_1;
	} else if (j <= 1.12e+73) {
		tmp = z * (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2.45e+17:
		tmp = t_1
	elif j <= 1.12e+73:
		tmp = z * (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2.45e+17)
		tmp = t_1;
	elseif (j <= 1.12e+73)
		tmp = Float64(z * Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2.45e+17)
		tmp = t_1;
	elseif (j <= 1.12e+73)
		tmp = z * (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.45e+17], t$95$1, If[LessEqual[j, 1.12e+73], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.45e17 or 1.12e73 < j

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(c \cdot b\right)} \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \]

      14. lower-neg.f64 54.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right)\right) \]

   8. Applied rewrites 54.6%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)}\right) \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. lower-*.f64 42.9

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   11. Applied rewrites 42.9%

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

   if -2.45e17 < j < 1.12e73

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 54.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. lower-*.f64 34.0

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Applied rewrites 34.0%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 1.12 \cdot 10^{+73}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 30.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.12 \cdot 10^{+73}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2.45e+17) t_1 (if (<= j 1.12e+73) (* y (* x z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.45e+17) {
		tmp = t_1;
	} else if (j <= 1.12e+73) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2.45d+17)) then
        tmp = t_1
    else if (j <= 1.12d+73) then
        tmp = y * (x * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2.45e+17) {
		tmp = t_1;
	} else if (j <= 1.12e+73) {
		tmp = y * (x * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2.45e+17:
		tmp = t_1
	elif j <= 1.12e+73:
		tmp = y * (x * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2.45e+17)
		tmp = t_1;
	elseif (j <= 1.12e+73)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2.45e+17)
		tmp = t_1;
	elseif (j <= 1.12e+73)
		tmp = y * (x * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.45e+17], t$95$1, If[LessEqual[j, 1.12e+73], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.45e17 or 1.12e73 < j

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(c \cdot b\right)} \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \]

      14. lower-neg.f64 54.6

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right)\right) \]

   8. Applied rewrites 54.6%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)}\right) \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. lower-*.f64 42.9

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   11. Applied rewrites 42.9%

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

   if -2.45e17 < j < 1.12e73

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. lower-*.f64 54.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto z \cdot \left(c \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + y \cdot x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto z \cdot \left(c \cdot \left(\mathsf{neg}\left(b\right)\right) + \color{blue}{y \cdot x}\right) \]

      3. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x + c \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto z \cdot \left(\color{blue}{y \cdot x} + c \cdot \left(\mathsf{neg}\left(b\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, c \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      6. lower-*.f64 54.6

         \[\leadsto z \cdot \mathsf{fma}\left(y, x, \color{blue}{c \cdot \left(-b\right)}\right) \]

   7. Applied rewrites 54.6%

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(y, x, c \cdot \left(-b\right)\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} \]

      6. lower-*.f64 33.3

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} \]

   10. Applied rewrites 33.3%

       \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 1.12 \cdot 10^{+73}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{+140}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= y -5.1e-11) t_1 (if (<= y 7.3e+140) (* c (* t j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -5.1e-11) {
		tmp = t_1;
	} else if (y <= 7.3e+140) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (y <= (-5.1d-11)) then
        tmp = t_1
    else if (y <= 7.3d+140) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -5.1e-11) {
		tmp = t_1;
	} else if (y <= 7.3e+140) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if y <= -5.1e-11:
		tmp = t_1
	elif y <= 7.3e+140:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -5.1e-11)
		tmp = t_1;
	elseif (y <= 7.3e+140)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (y <= -5.1e-11)
		tmp = t_1;
	elseif (y <= 7.3e+140)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.1e-11], t$95$1, If[LessEqual[y, 7.3e+140], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.09999999999999984e-11 or 7.3000000000000004e140 < y

   1. Initial program 65.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. lower-*.f64 55.4

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      2. lower-*.f64 46.6

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Applied rewrites 46.6%

      \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   if -5.09999999999999984e-11 < y < 7.3000000000000004e140

   1. Initial program 77.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(c \cdot b\right)} \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \]

      14. lower-neg.f64 62.2

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right)\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)}\right) \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. lower-*.f64 31.0

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   11. Applied rewrites 31.0%

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{+140}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;i \leq -1.35 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+37}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= i -1.35e+110) t_1 (if (<= i 4e+37) (* j (* t c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -1.35e+110) {
		tmp = t_1;
	} else if (i <= 4e+37) {
		tmp = j * (t * 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 = a * (b * i)
    if (i <= (-1.35d+110)) then
        tmp = t_1
    else if (i <= 4d+37) then
        tmp = j * (t * 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 = a * (b * i);
	double tmp;
	if (i <= -1.35e+110) {
		tmp = t_1;
	} else if (i <= 4e+37) {
		tmp = j * (t * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if i <= -1.35e+110:
		tmp = t_1
	elif i <= 4e+37:
		tmp = j * (t * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(b * i))
	tmp = 0.0
	if (i <= -1.35e+110)
		tmp = t_1;
	elseif (i <= 4e+37)
		tmp = Float64(j * Float64(t * 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 = a * (b * i);
	tmp = 0.0;
	if (i <= -1.35e+110)
		tmp = t_1;
	elseif (i <= 4e+37)
		tmp = j * (t * 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[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.35e+110], t$95$1, If[LessEqual[i, 4e+37], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.35000000000000005e110 or 3.99999999999999982e37 < i

   1. Initial program 62.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 40.4

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

      2. lower-*.f64 33.8

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

   8. Applied rewrites 33.8%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   if -1.35000000000000005e110 < i < 3.99999999999999982e37

   1. Initial program 80.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      6. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      9. lower-neg.f64 41.8

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, y \cdot \color{blue}{\left(-i\right)}\right) \]

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, y \cdot \left(-i\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot j\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot c\right)} \cdot t \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t\right)} \]

      5. lower-*.f64 33.2

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

   8. Applied rewrites 33.2%

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.35 \cdot 10^{+110}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+37}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 30.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(b \cdot i\right)\\ \mathbf{if}\;i \leq -2.25 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* b i))))
   (if (<= i -2.25e+117) t_1 (if (<= i 3.6e+37) (* c (* t j)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -2.25e+117) {
		tmp = t_1;
	} else if (i <= 3.6e+37) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (b * i)
    if (i <= (-2.25d+117)) then
        tmp = t_1
    else if (i <= 3.6d+37) then
        tmp = c * (t * j)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (b * i);
	double tmp;
	if (i <= -2.25e+117) {
		tmp = t_1;
	} else if (i <= 3.6e+37) {
		tmp = c * (t * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (b * i)
	tmp = 0
	if i <= -2.25e+117:
		tmp = t_1
	elif i <= 3.6e+37:
		tmp = c * (t * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(b * i))
	tmp = 0.0
	if (i <= -2.25e+117)
		tmp = t_1;
	elseif (i <= 3.6e+37)
		tmp = Float64(c * Float64(t * j));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (b * i);
	tmp = 0.0;
	if (i <= -2.25e+117)
		tmp = t_1;
	elseif (i <= 3.6e+37)
		tmp = c * (t * j);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.25e+117], t$95$1, If[LessEqual[i, 3.6e+37], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.25e117 or 3.59999999999999998e37 < i

   1. Initial program 63.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 41.5

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

      2. lower-*.f64 34.6

         \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   if -2.25e117 < i < 3.59999999999999998e37

   1. Initial program 80.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]

   5. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, z \cdot x\right), b \cdot \mathsf{fma}\left(c, -z, i \cdot a\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(b \cdot c\right) \cdot z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{\left(c \cdot b\right)} \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{neg}\left(\color{blue}{c \cdot \left(b \cdot z\right)}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)}\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(\mathsf{neg}\left(x\right)\right)\right), c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \]

      14. lower-neg.f64 60.0

          \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), c \cdot \left(z \cdot \color{blue}{\left(-b\right)}\right)\right) \]

   8. Applied rewrites 60.0%

      \[\leadsto \mathsf{fma}\left(t, \mathsf{fma}\left(j, c, a \cdot \left(-x\right)\right), \color{blue}{c \cdot \left(z \cdot \left(-b\right)\right)}\right) \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

       2. lower-*.f64 32.4

          \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   11. Applied rewrites 32.4%

       \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.25 \cdot 10^{+117}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+37}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 22.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* b i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * 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 = a * (b * 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 a * (b * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (b * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(b * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (b * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

   12. lower-*.f64 32.4

       \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

5. Applied rewrites 32.4%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   2. lower-*.f64 18.9

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

8. Applied rewrites 18.9%

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
9. Add Preprocessing

Developer Target 1: 68.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))