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

Percentage Accurate: 74.3% → 82.7%

Time: 16.9s

Alternatives: 23

Speedup: 0.9×

• 58.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 90.8

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 90.8

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

   4. Applied rewrites 90.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 82.8%

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

Alternative 2: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{if}\;j \leq -4.3 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 4.7 \cdot 10^{+79}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot t - c \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- i) y (* c a)) j (* (fma (- t) a (* z y)) x))))
   (if (<= j -4.3e+30)
     t_1
     (if (<= j 4.7e+79)
       (-
        (* (* j c) a)
        (- (* (- (* a t) (* z y)) x) (* (- (* i t) (* c z)) b)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-i, y, (c * a)), j, (fma(-t, a, (z * y)) * x));
	double tmp;
	if (j <= -4.3e+30) {
		tmp = t_1;
	} else if (j <= 4.7e+79) {
		tmp = ((j * c) * a) - ((((a * t) - (z * y)) * x) - (((i * t) - (c * z)) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-t), a, Float64(z * y)) * x))
	tmp = 0.0
	if (j <= -4.3e+30)
		tmp = t_1;
	elseif (j <= 4.7e+79)
		tmp = Float64(Float64(Float64(j * c) * a) - Float64(Float64(Float64(Float64(a * t) - Float64(z * y)) * x) - Float64(Float64(Float64(i * t) - Float64(c * z)) * b)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -4.3e+30], t$95$1, If[LessEqual[j, 4.7e+79], N[(N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision] - N[(N[(N[(N[(a * t), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -4.3e30 or 4.70000000000000023e79 < j

   1. Initial program 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 71.4

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 73.3

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   if -4.3e30 < j < 4.70000000000000023e79

   1. Initial program 72.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -4.3 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;j \leq 4.7 \cdot 10^{+79}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a - \left(\left(a \cdot t - z \cdot y\right) \cdot x - \left(i \cdot t - c \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x - \left(i \cdot y - c \cdot a\right) \cdot j\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -1.3e+53)
     t_1
     (if (<= b -4e-45)
       (- (* (* (- t) a) x) (* (- (* i y) (* c a)) j))
       (if (<= b 5.3e+129) (+ (* (* z x) y) (* (- (* c a) (* i y)) 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 = fma(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -1.3e+53) {
		tmp = t_1;
	} else if (b <= -4e-45) {
		tmp = ((-t * a) * x) - (((i * y) - (c * a)) * j);
	} else if (b <= 5.3e+129) {
		tmp = ((z * x) * y) + (((c * a) - (i * y)) * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -1.3e+53)
		tmp = t_1;
	elseif (b <= -4e-45)
		tmp = Float64(Float64(Float64(Float64(-t) * a) * x) - Float64(Float64(Float64(i * y) - Float64(c * a)) * j));
	elseif (b <= 5.3e+129)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(Float64(Float64(c * a) - Float64(i * y)) * 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[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.3e+53], t$95$1, If[LessEqual[b, -4e-45], N[(N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(i * y), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.3e+129], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.29999999999999999e53 or 5.2999999999999999e129 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.29999999999999999e53 < b < -3.99999999999999994e-45

   1. Initial program 80.9%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -3.99999999999999994e-45 < b < 5.2999999999999999e129

   1. Initial program 73.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x - \left(i \cdot y - c \cdot a\right) \cdot j\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -5.3 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -5.3e+53)
     t_1
     (if (<= b 2.4e+131)
       (fma (fma (- i) y (* c a)) j (* (fma (- t) a (* z y)) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -5.3e+53) {
		tmp = t_1;
	} else if (b <= 2.4e+131) {
		tmp = fma(fma(-i, y, (c * a)), j, (fma(-t, a, (z * y)) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -5.3e+53)
		tmp = t_1;
	elseif (b <= 2.4e+131)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -5.3e+53], t$95$1, If[LessEqual[b, 2.4e+131], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.3000000000000002e53 or 2.3999999999999999e131 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -5.3000000000000002e53 < b < 2.3999999999999999e131

   1. Initial program 74.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.2

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 76.4

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 74.8

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

   7. Applied rewrites 74.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.65 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -1.65e+53)
     t_1
     (if (<= b 2.4e+131)
       (fma (fma (- t) a (* z y)) x (* (fma (- i) y (* c a)) j))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -1.65e+53) {
		tmp = t_1;
	} else if (b <= 2.4e+131) {
		tmp = fma(fma(-t, a, (z * y)), x, (fma(-i, y, (c * a)) * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -1.65e+53)
		tmp = t_1;
	elseif (b <= 2.4e+131)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-i), y, Float64(c * a)) * 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[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.65e+53], t$95$1, If[LessEqual[b, 2.4e+131], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6500000000000001e53 or 2.3999999999999999e131 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.6500000000000001e53 < b < 2.3999999999999999e131

   1. Initial program 74.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

Alternative 6: 28.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -2.5e+113)
     t_1
     (if (<= z -5.5e-178)
       (* (* j a) c)
       (if (<= z -1.28e-250)
         (* (* (- t) a) x)
         (if (<= z 6.8e-207)
           (* (* j c) a)
           (if (<= z 1.8e-25)
             (* (* i b) t)
             (if (<= z 6.5e+108) t_1 (* (* (- c) z) b)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= -5.5e-178) {
		tmp = (j * a) * c;
	} else if (z <= -1.28e-250) {
		tmp = (-t * a) * x;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 6.5e+108) {
		tmp = t_1;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-2.5d+113)) then
        tmp = t_1
    else if (z <= (-5.5d-178)) then
        tmp = (j * a) * c
    else if (z <= (-1.28d-250)) then
        tmp = (-t * a) * x
    else if (z <= 6.8d-207) then
        tmp = (j * c) * a
    else if (z <= 1.8d-25) then
        tmp = (i * b) * t
    else if (z <= 6.5d+108) then
        tmp = t_1
    else
        tmp = (-c * z) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= -5.5e-178) {
		tmp = (j * a) * c;
	} else if (z <= -1.28e-250) {
		tmp = (-t * a) * x;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 6.5e+108) {
		tmp = t_1;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if z <= -2.5e+113:
		tmp = t_1
	elif z <= -5.5e-178:
		tmp = (j * a) * c
	elif z <= -1.28e-250:
		tmp = (-t * a) * x
	elif z <= 6.8e-207:
		tmp = (j * c) * a
	elif z <= 1.8e-25:
		tmp = (i * b) * t
	elif z <= 6.5e+108:
		tmp = t_1
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= -5.5e-178)
		tmp = Float64(Float64(j * a) * c);
	elseif (z <= -1.28e-250)
		tmp = Float64(Float64(Float64(-t) * a) * x);
	elseif (z <= 6.8e-207)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 1.8e-25)
		tmp = Float64(Float64(i * b) * t);
	elseif (z <= 6.5e+108)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-c) * z) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= -5.5e-178)
		tmp = (j * a) * c;
	elseif (z <= -1.28e-250)
		tmp = (-t * a) * x;
	elseif (z <= 6.8e-207)
		tmp = (j * c) * a;
	elseif (z <= 1.8e-25)
		tmp = (i * b) * t;
	elseif (z <= 6.5e+108)
		tmp = t_1;
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -2.5e+113], t$95$1, If[LessEqual[z, -5.5e-178], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, -1.28e-250], N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.8e-207], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.8e-25], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 6.5e+108], t$95$1, N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -2.5e113 or 1.8e-25 < z < 6.4999999999999996e108

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 50.8%

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

   if -2.5e113 < z < -5.50000000000000028e-178

   1. Initial program 74.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if -5.50000000000000028e-178 < z < -1.27999999999999993e-250

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.1%

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

   if -1.27999999999999993e-250 < z < 6.79999999999999997e-207

   1. Initial program 83.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 37.4%

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

   if 6.79999999999999997e-207 < z < 1.8e-25

   1. Initial program 78.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 6.4999999999999996e108 < z

   1. Initial program 58.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.5%

      \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot b \]
3. Recombined 6 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+108}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot b\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -2.5e+113)
     t_1
     (if (<= z -5.5e-178)
       (* (* j a) c)
       (if (<= z -1.28e-250)
         (* (* (- t) a) x)
         (if (<= z 6.8e-207)
           (* (* j c) a)
           (if (<= z 1.8e-25)
             (* (* i b) t)
             (if (<= z 1.2e+111) t_1 (* (* (- c) b) 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 * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= -5.5e-178) {
		tmp = (j * a) * c;
	} else if (z <= -1.28e-250) {
		tmp = (-t * a) * x;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 1.2e+111) {
		tmp = t_1;
	} else {
		tmp = (-c * b) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-2.5d+113)) then
        tmp = t_1
    else if (z <= (-5.5d-178)) then
        tmp = (j * a) * c
    else if (z <= (-1.28d-250)) then
        tmp = (-t * a) * x
    else if (z <= 6.8d-207) then
        tmp = (j * c) * a
    else if (z <= 1.8d-25) then
        tmp = (i * b) * t
    else if (z <= 1.2d+111) then
        tmp = t_1
    else
        tmp = (-c * b) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= -5.5e-178) {
		tmp = (j * a) * c;
	} else if (z <= -1.28e-250) {
		tmp = (-t * a) * x;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 1.2e+111) {
		tmp = t_1;
	} else {
		tmp = (-c * b) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if z <= -2.5e+113:
		tmp = t_1
	elif z <= -5.5e-178:
		tmp = (j * a) * c
	elif z <= -1.28e-250:
		tmp = (-t * a) * x
	elif z <= 6.8e-207:
		tmp = (j * c) * a
	elif z <= 1.8e-25:
		tmp = (i * b) * t
	elif z <= 1.2e+111:
		tmp = t_1
	else:
		tmp = (-c * b) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= -5.5e-178)
		tmp = Float64(Float64(j * a) * c);
	elseif (z <= -1.28e-250)
		tmp = Float64(Float64(Float64(-t) * a) * x);
	elseif (z <= 6.8e-207)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 1.8e-25)
		tmp = Float64(Float64(i * b) * t);
	elseif (z <= 1.2e+111)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-c) * b) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= -5.5e-178)
		tmp = (j * a) * c;
	elseif (z <= -1.28e-250)
		tmp = (-t * a) * x;
	elseif (z <= 6.8e-207)
		tmp = (j * c) * a;
	elseif (z <= 1.8e-25)
		tmp = (i * b) * t;
	elseif (z <= 1.2e+111)
		tmp = t_1;
	else
		tmp = (-c * b) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -2.5e+113], t$95$1, If[LessEqual[z, -5.5e-178], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, -1.28e-250], N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.8e-207], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.8e-25], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 1.2e+111], t$95$1, N[(N[((-c) * b), $MachinePrecision] * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -2.5e113 or 1.8e-25 < z < 1.20000000000000003e111

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 50.8%

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

   if -2.5e113 < z < -5.50000000000000028e-178

   1. Initial program 74.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.8%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if -5.50000000000000028e-178 < z < -1.27999999999999993e-250

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.1%

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

   if -1.27999999999999993e-250 < z < 6.79999999999999997e-207

   1. Initial program 83.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 37.4%

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

   if 6.79999999999999997e-207 < z < 1.8e-25

   1. Initial program 78.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 1.20000000000000003e111 < z

   1. Initial program 58.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.2%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 6 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot b\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-32}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -2.7e-49)
     t_1
     (if (<= z -5.5e-178)
       (* (* j a) c)
       (if (<= z -1.28e-250)
         (* (* (- t) a) x)
         (if (<= z 6.8e-207)
           (* (* j c) a)
           (if (<= z 4.1e-32) (* (* i b) t) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -2.7e-49) {
		tmp = t_1;
	} else if (z <= -5.5e-178) {
		tmp = (j * a) * c;
	} else if (z <= -1.28e-250) {
		tmp = (-t * a) * x;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 4.1e-32) {
		tmp = (i * b) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -2.7e-49)
		tmp = t_1;
	elseif (z <= -5.5e-178)
		tmp = Float64(Float64(j * a) * c);
	elseif (z <= -1.28e-250)
		tmp = Float64(Float64(Float64(-t) * a) * x);
	elseif (z <= 6.8e-207)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 4.1e-32)
		tmp = Float64(Float64(i * b) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.7e-49], t$95$1, If[LessEqual[z, -5.5e-178], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, -1.28e-250], N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.8e-207], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 4.1e-32], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.7e-49 or 4.09999999999999975e-32 < z

   1. Initial program 64.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -2.7e-49 < z < -5.50000000000000028e-178

   1. Initial program 78.5%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 39.7

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.7%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if -5.50000000000000028e-178 < z < -1.27999999999999993e-250

   1. Initial program 62.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.1%

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

   if -1.27999999999999993e-250 < z < 6.79999999999999997e-207

   1. Initial program 83.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 37.4%

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

   if 6.79999999999999997e-207 < z < 4.09999999999999975e-32

   1. Initial program 78.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-32}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -1.2e+49)
     t_1
     (if (<= b 5.3e+129) (+ (* (* z x) y) (* (- (* c a) (* i y)) 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 = fma(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -1.2e+49) {
		tmp = t_1;
	} else if (b <= 5.3e+129) {
		tmp = ((z * x) * y) + (((c * a) - (i * y)) * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -1.2e+49)
		tmp = t_1;
	elseif (b <= 5.3e+129)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(Float64(Float64(c * a) - Float64(i * y)) * 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[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.2e+49], t$95$1, If[LessEqual[b, 5.3e+129], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.2e49 or 5.2999999999999999e129 < b

   1. Initial program 63.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if -1.2e49 < b < 5.2999999999999999e129

   1. Initial program 73.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.45e+113)
   (* (fma (- i) j (* z x)) y)
   (if (<= z -4.4e+45)
     (* (fma (- z) b (* j a)) c)
     (if (<= z 6e-223)
       (* (fma (- i) y (* c a)) j)
       (if (<= z 9.6e+56)
         (* (fma (- x) a (* i b)) t)
         (* (fma (- c) b (* y x)) z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.45e+113) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= -4.4e+45) {
		tmp = fma(-z, b, (j * a)) * c;
	} else if (z <= 6e-223) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (z <= 9.6e+56) {
		tmp = fma(-x, a, (i * b)) * t;
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.45e+113)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= -4.4e+45)
		tmp = Float64(fma(Float64(-z), b, Float64(j * a)) * c);
	elseif (z <= 6e-223)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (z <= 9.6e+56)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.45e+113], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -4.4e+45], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, 6e-223], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 9.6e+56], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.45000000000000011e113

   1. Initial program 55.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -2.45000000000000011e113 < z < -4.4000000000000001e45

   1. Initial program 78.9%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if -4.4000000000000001e45 < z < 5.99999999999999983e-223

   1. Initial program 75.8%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 51.5

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

   5. Applied rewrites 51.5%

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

   if 5.99999999999999983e-223 < z < 9.60000000000000053e56

   1. Initial program 77.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if 9.60000000000000053e56 < z

   1. Initial program 60.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 48.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.3e-47)
   (* (fma (- i) j (* z x)) y)
   (if (<= z -9e-124)
     (* (fma (- y) j (* b t)) i)
     (if (<= z 6e-223)
       (* (fma (- i) y (* c a)) j)
       (if (<= z 9.6e+56)
         (* (fma (- x) a (* i b)) t)
         (* (fma (- c) b (* y x)) z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.3e-47) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= -9e-124) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 6e-223) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (z <= 9.6e+56) {
		tmp = fma(-x, a, (i * b)) * t;
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.3e-47)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= -9e-124)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 6e-223)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (z <= 9.6e+56)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.3e-47], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, -9e-124], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 6e-223], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 9.6e+56], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.3e-47

   1. Initial program 62.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   if -1.3e-47 < z < -8.9999999999999992e-124

   1. Initial program 74.8%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -8.9999999999999992e-124 < z < 5.99999999999999983e-223

   1. Initial program 79.9%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   if 5.99999999999999983e-223 < z < 9.60000000000000053e56

   1. Initial program 77.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if 9.60000000000000053e56 < z

   1. Initial program 60.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 28.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot b\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -2.5e+113)
     t_1
     (if (<= z 6.6e-207)
       (* (* j a) c)
       (if (<= z 1.8e-25)
         (* (* i b) t)
         (if (<= z 1.2e+111) t_1 (* (* (- c) b) 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 * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= 6.6e-207) {
		tmp = (j * a) * c;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 1.2e+111) {
		tmp = t_1;
	} else {
		tmp = (-c * b) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-2.5d+113)) then
        tmp = t_1
    else if (z <= 6.6d-207) then
        tmp = (j * a) * c
    else if (z <= 1.8d-25) then
        tmp = (i * b) * t
    else if (z <= 1.2d+111) then
        tmp = t_1
    else
        tmp = (-c * b) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= 6.6e-207) {
		tmp = (j * a) * c;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 1.2e+111) {
		tmp = t_1;
	} else {
		tmp = (-c * b) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if z <= -2.5e+113:
		tmp = t_1
	elif z <= 6.6e-207:
		tmp = (j * a) * c
	elif z <= 1.8e-25:
		tmp = (i * b) * t
	elif z <= 1.2e+111:
		tmp = t_1
	else:
		tmp = (-c * b) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= 6.6e-207)
		tmp = Float64(Float64(j * a) * c);
	elseif (z <= 1.8e-25)
		tmp = Float64(Float64(i * b) * t);
	elseif (z <= 1.2e+111)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-c) * b) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= 6.6e-207)
		tmp = (j * a) * c;
	elseif (z <= 1.8e-25)
		tmp = (i * b) * t;
	elseif (z <= 1.2e+111)
		tmp = t_1;
	else
		tmp = (-c * b) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -2.5e+113], t$95$1, If[LessEqual[z, 6.6e-207], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, 1.8e-25], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 1.2e+111], t$95$1, N[(N[((-c) * b), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.5e113 or 1.8e-25 < z < 1.20000000000000003e111

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.9%

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

   10. Applied rewrites 50.8%

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

   if -2.5e113 < z < 6.5999999999999996e-207

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.4%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if 6.5999999999999996e-207 < z < 1.8e-25

   1. Initial program 78.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 1.20000000000000003e111 < z

   1. Initial program 58.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.2%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot b\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -2.5e+113)
     t_1
     (if (<= z 6.6e-207)
       (* (* j a) c)
       (if (<= z 1.8e-25)
         (* (* i b) t)
         (if (<= z 1.2e+111) t_1 (* (* (- z) b) c)))))))

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 (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= 6.6e-207) {
		tmp = (j * a) * c;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 1.2e+111) {
		tmp = t_1;
	} else {
		tmp = (-z * b) * c;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-2.5d+113)) then
        tmp = t_1
    else if (z <= 6.6d-207) then
        tmp = (j * a) * c
    else if (z <= 1.8d-25) then
        tmp = (i * b) * t
    else if (z <= 1.2d+111) then
        tmp = t_1
    else
        tmp = (-z * b) * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -2.5e+113) {
		tmp = t_1;
	} else if (z <= 6.6e-207) {
		tmp = (j * a) * c;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else if (z <= 1.2e+111) {
		tmp = t_1;
	} else {
		tmp = (-z * b) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if z <= -2.5e+113:
		tmp = t_1
	elif z <= 6.6e-207:
		tmp = (j * a) * c
	elif z <= 1.8e-25:
		tmp = (i * b) * t
	elif z <= 1.2e+111:
		tmp = t_1
	else:
		tmp = (-z * b) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= 6.6e-207)
		tmp = Float64(Float64(j * a) * c);
	elseif (z <= 1.8e-25)
		tmp = Float64(Float64(i * b) * t);
	elseif (z <= 1.2e+111)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-z) * b) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -2.5e+113)
		tmp = t_1;
	elseif (z <= 6.6e-207)
		tmp = (j * a) * c;
	elseif (z <= 1.8e-25)
		tmp = (i * b) * t;
	elseif (z <= 1.2e+111)
		tmp = t_1;
	else
		tmp = (-z * b) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -2.5e+113], t$95$1, If[LessEqual[z, 6.6e-207], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, 1.8e-25], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 1.2e+111], t$95$1, N[(N[((-z) * b), $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.5e113 or 1.8e-25 < z < 1.20000000000000003e111

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.9%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.8%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -2.5e113 < z < 6.5999999999999996e-207

   1. Initial program 75.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 38.2

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if 6.5999999999999996e-207 < z < 1.8e-25

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot t \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \color{blue}{b \cdot i}\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

      12. lower-*.f64 56.3

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 1.20000000000000003e111 < z

   1. Initial program 58.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 82.9

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.5%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.1%

       \[\leadsto \left(\left(-b\right) \cdot z\right) \cdot \color{blue}{c} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -3.6e-12)
   (* (fma (- i) j (* z x)) y)
   (if (<= z 6e-223)
     (* (fma (- i) y (* c a)) j)
     (if (<= z 9.6e+56)
       (* (fma (- x) a (* i b)) t)
       (* (fma (- c) b (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -3.6e-12) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= 6e-223) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (z <= 9.6e+56) {
		tmp = fma(-x, a, (i * b)) * t;
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -3.6e-12)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= 6e-223)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (z <= 9.6e+56)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -3.6e-12], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 6e-223], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 9.6e+56], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.6e-12

   1. Initial program 61.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, j, x \cdot z\right)} \cdot y \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, j, x \cdot z\right) \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 57.6

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   if -3.6e-12 < z < 5.99999999999999983e-223

   1. Initial program 77.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

      10. lower-*.f64 52.7

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j} \]

   if 5.99999999999999983e-223 < z < 9.60000000000000053e56

   1. Initial program 77.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot t \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \color{blue}{b \cdot i}\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

      12. lower-*.f64 56.4

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t} \]

   if 9.60000000000000053e56 < z

   1. Initial program 60.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 81.3

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -1.05e+30)
     t_1
     (if (<= x 6.5e-23) (* (fma (- i) y (* c a)) j) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -1.05e+30) {
		tmp = t_1;
	} else if (x <= 6.5e-23) {
		tmp = fma(-i, y, (c * a)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -1.05e+30)
		tmp = t_1;
	elseif (x <= 6.5e-23)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * 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[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.05e+30], t$95$1, If[LessEqual[x, 6.5e-23], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e30 or 6.5e-23 < x

   1. Initial program 68.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{t \cdot a}\right)\right) + y \cdot z\right) \cdot x \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot a} + y \cdot z\right) \cdot x \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right)} \cdot a + y \cdot z\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, a, y \cdot z\right)} \cdot x \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, a, y \cdot z\right) \cdot x \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-t}, a, y \cdot z\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-t, a, \color{blue}{z \cdot y}\right) \cdot x \]

      12. lower-*.f64 64.4

          \[\leadsto \mathsf{fma}\left(-t, a, \color{blue}{z \cdot y}\right) \cdot x \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x} \]

   if -1.05e30 < x < 6.5e-23

   1. Initial program 72.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

      10. lower-*.f64 49.8

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 48.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -3.6e-12)
   (* (fma (- i) j (* z x)) y)
   (if (<= z 1.8e-13)
     (* (fma (- i) y (* c a)) j)
     (* (fma (- c) b (* y x)) z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -3.6e-12) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (z <= 1.8e-13) {
		tmp = fma(-i, y, (c * a)) * j;
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -3.6e-12)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (z <= 1.8e-13)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -3.6e-12], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.8e-13], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6e-12

   1. Initial program 61.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, j, x \cdot z\right)} \cdot y \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, j, x \cdot z\right) \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 57.6

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   if -3.6e-12 < z < 1.7999999999999999e-13

   1. Initial program 77.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c\right)} \cdot j \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right)} \cdot y + a \cdot c\right) \cdot j \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)} \cdot j \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right) \cdot j \]

      8. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

      10. lower-*.f64 48.3

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right) \cdot j \]

   5. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j} \]

   if 1.7999999999999999e-13 < z

   1. Initial program 63.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 73.3

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 28.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.5e+113)
   (* (* z x) y)
   (if (<= z 6.6e-207)
     (* (* j a) c)
     (if (<= z 1.8e-25) (* (* i b) t) (* (* y x) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.5e+113) {
		tmp = (z * x) * y;
	} else if (z <= 6.6e-207) {
		tmp = (j * a) * c;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-2.5d+113)) then
        tmp = (z * x) * y
    else if (z <= 6.6d-207) then
        tmp = (j * a) * c
    else if (z <= 1.8d-25) then
        tmp = (i * b) * t
    else
        tmp = (y * x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -2.5e+113) {
		tmp = (z * x) * y;
	} else if (z <= 6.6e-207) {
		tmp = (j * a) * c;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.5e+113:
		tmp = (z * x) * y
	elif z <= 6.6e-207:
		tmp = (j * a) * c
	elif z <= 1.8e-25:
		tmp = (i * b) * t
	else:
		tmp = (y * x) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.5e+113)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= 6.6e-207)
		tmp = Float64(Float64(j * a) * c);
	elseif (z <= 1.8e-25)
		tmp = Float64(Float64(i * b) * t);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -2.5e+113)
		tmp = (z * x) * y;
	elseif (z <= 6.6e-207)
		tmp = (j * a) * c;
	elseif (z <= 1.8e-25)
		tmp = (i * b) * t;
	else
		tmp = (y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.5e+113], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 6.6e-207], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[z, 1.8e-25], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.5e113

   1. Initial program 55.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 63.1

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.7%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.0%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -2.5e113 < z < 6.5999999999999996e-207

   1. Initial program 75.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 38.2

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if 6.5999999999999996e-207 < z < 1.8e-25

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot t \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \color{blue}{b \cdot i}\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

      12. lower-*.f64 56.3

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 1.8e-25 < z

   1. Initial program 65.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 41.9%

      \[\leadsto \left(x \cdot y\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 28.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-19}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.15e-19)
   (* (* z x) y)
   (if (<= z 6.8e-207)
     (* (* j c) a)
     (if (<= z 1.8e-25) (* (* i b) t) (* (* y x) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.15e-19) {
		tmp = (z * x) * y;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-1.15d-19)) then
        tmp = (z * x) * y
    else if (z <= 6.8d-207) then
        tmp = (j * c) * a
    else if (z <= 1.8d-25) then
        tmp = (i * b) * t
    else
        tmp = (y * x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.15e-19) {
		tmp = (z * x) * y;
	} else if (z <= 6.8e-207) {
		tmp = (j * c) * a;
	} else if (z <= 1.8e-25) {
		tmp = (i * b) * t;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -1.15e-19:
		tmp = (z * x) * y
	elif z <= 6.8e-207:
		tmp = (j * c) * a
	elif z <= 1.8e-25:
		tmp = (i * b) * t
	else:
		tmp = (y * x) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.15e-19)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= 6.8e-207)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 1.8e-25)
		tmp = Float64(Float64(i * b) * t);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -1.15e-19)
		tmp = (z * x) * y;
	elseif (z <= 6.8e-207)
		tmp = (j * c) * a;
	elseif (z <= 1.8e-25)
		tmp = (i * b) * t;
	else
		tmp = (y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.15e-19], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 6.8e-207], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 1.8e-25], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1499999999999999e-19

   1. Initial program 61.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.8%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 41.6%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -1.1499999999999999e-19 < z < 6.79999999999999997e-207

   1. Initial program 76.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 34.0

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 34.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]

   if 6.79999999999999997e-207 < z < 1.8e-25

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot t \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot t \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot a + \color{blue}{b \cdot i}\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

      12. lower-*.f64 56.3

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 40.2%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 1.8e-25 < z

   1. Initial program 65.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 41.9%

      \[\leadsto \left(x \cdot y\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-19}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-207}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z 2.1e+89) (* (fma (- i) j (* z x)) y) (* (fma (- c) b (* y x)) z)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= 2.1e+89) {
		tmp = fma(-i, j, (z * x)) * y;
	} else {
		tmp = fma(-c, b, (y * x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= 2.1e+89)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, 2.1e+89], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.09999999999999986e89

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, j, x \cdot z\right)} \cdot y \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, j, x \cdot z\right) \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 42.1

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   if 2.09999999999999986e89 < z

   1. Initial program 57.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 84.0

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-39}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-41}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -2.1e-39)
   (* (* y x) z)
   (if (<= x 2.25e-41) (* (* j c) a) (* (* z x) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -2.1e-39) {
		tmp = (y * x) * z;
	} else if (x <= 2.25e-41) {
		tmp = (j * c) * a;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= (-2.1d-39)) then
        tmp = (y * x) * z
    else if (x <= 2.25d-41) then
        tmp = (j * c) * a
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -2.1e-39) {
		tmp = (y * x) * z;
	} else if (x <= 2.25e-41) {
		tmp = (j * c) * a;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -2.1e-39:
		tmp = (y * x) * z
	elif x <= 2.25e-41:
		tmp = (j * c) * a
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -2.1e-39)
		tmp = Float64(Float64(y * x) * z);
	elseif (x <= 2.25e-41)
		tmp = Float64(Float64(j * c) * a);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= -2.1e-39)
		tmp = (y * x) * z;
	elseif (x <= 2.25e-41)
		tmp = (j * c) * a;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -2.1e-39], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 2.25e-41], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999993e-39

   1. Initial program 78.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 42.7

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 33.7%

      \[\leadsto \left(x \cdot y\right) \cdot z \]

   if -2.09999999999999993e-39 < x < 2.25e-41

   1. Initial program 71.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 51.1

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.4%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]

   if 2.25e-41 < x

   1. Initial program 59.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 44.9

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.7%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.2%

       \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-39}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-41}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 28.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-49}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -1.15e-19) t_1 (if (<= z 7.2e-49) (* (* j c) 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 * x) * y;
	double tmp;
	if (z <= -1.15e-19) {
		tmp = t_1;
	} else if (z <= 7.2e-49) {
		tmp = (j * c) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-1.15d-19)) then
        tmp = t_1
    else if (z <= 7.2d-49) then
        tmp = (j * c) * a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -1.15e-19) {
		tmp = t_1;
	} else if (z <= 7.2e-49) {
		tmp = (j * c) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if z <= -1.15e-19:
		tmp = t_1
	elif z <= 7.2e-49:
		tmp = (j * c) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -1.15e-19)
		tmp = t_1;
	elseif (z <= 7.2e-49)
		tmp = Float64(Float64(j * c) * a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -1.15e-19)
		tmp = t_1;
	elseif (z <= 7.2e-49)
		tmp = (j * c) * a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.15e-19], t$95$1, If[LessEqual[z, 7.2e-49], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1499999999999999e-19 or 7.19999999999999939e-49 < z

   1. Initial program 63.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 59.3

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.0%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.0%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -1.1499999999999999e-19 < z < 7.19999999999999939e-49

   1. Initial program 77.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + a \cdot j\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + a \cdot j\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + a \cdot j\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, a \cdot j\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, a \cdot j\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, a \cdot j\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

      12. lower-*.f64 36.3

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot a}\right) \cdot c \]

   5. Applied rewrites 36.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.3%

      \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 21.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot x\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z x) y))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * 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 = (z * x) * 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 (z * x) * y;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * x) * y

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * x) * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * x) * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. *-commutative N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

   9. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   12. lower-*.f64 39.1

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 39.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 23.4%

   \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 24.4%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
11. Add Preprocessing

Alternative 23: 22.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z y) x))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (z * y) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * y) * x

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. *-commutative N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

   9. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   12. lower-*.f64 39.1

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 39.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 23.4%

   \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]

9. Final simplification 23.4%

   \[\leadsto \left(z \cdot y\right) \cdot x \]
10. Add Preprocessing

Developer Target 1: 59.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024283 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))