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

Percentage Accurate: 73.5% → 84.6%

Time: 15.5s

Alternatives: 26

Speedup: 0.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% 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: 84.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (-
          (* (- (* c a) (* i y)) j)
          (- (* (- (* a t) (* z y)) x) (* (- (* i t) (* c z)) b)))))
   (if (<= t_1 INFINITY)
     t_1
     (* (fma j a (/ (fma (- a) (* t x) (* (fma (- i) j (* z x)) y)) c)) 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 = (((c * a) - (i * y)) * j) - ((((a * t) - (z * y)) * x) - (((i * t) - (c * z)) * b));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(j, a, (fma(-a, (t * x), (fma(-i, j, (z * x)) * y)) / c)) * c;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(j * a + N[(N[((-a) * N[(t * x), $MachinePrecision] + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $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 92.7%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in c around -inf

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

   10. Applied rewrites 66.0%

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

4. Final simplification 88.4%

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

Alternative 2: 75.0% accurate, 0.5× speedup

Math

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

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)
   (- (- (* (- (* z y) (* a t)) x) (* (* c b) z)) (* (- (* i y) (* c a)) j))
   (* (fma j a (/ (fma (- a) (* t x) (* (fma (- i) j (* z x)) y)) c)) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (((((c * a) - (i * y)) * j) - ((((a * t) - (z * y)) * x) - (((i * t) - (c * z)) * b))) <= ((double) INFINITY)) {
		tmp = ((((z * y) - (a * t)) * x) - ((c * b) * z)) - (((i * y) - (c * a)) * j);
	} else {
		tmp = fma(j, a, (fma(-a, (t * x), (fma(-i, j, (z * x)) * y)) / c)) * c;
	}
	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 = Float64(Float64(Float64(Float64(Float64(z * y) - Float64(a * t)) * x) - Float64(Float64(c * b) * z)) - Float64(Float64(Float64(i * y) - Float64(c * a)) * j));
	else
		tmp = Float64(fma(j, a, Float64(fma(Float64(-a), Float64(t * x), Float64(fma(Float64(-i), j, Float64(z * x)) * y)) / c)) * c);
	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[(N[(N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(c * b), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(i * y), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[(j * a + N[(N[((-a) * N[(t * x), $MachinePrecision] + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $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 92.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in c around -inf

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

   10. Applied rewrites 66.0%

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

4. Final simplification 77.7%

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

Alternative 3: 50.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+76}:\\ \;\;\;\;\left(\left(a - \frac{b \cdot z}{j}\right) \cdot c\right) \cdot j\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, j \cdot i\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -1.9e+177)
     t_1
     (if (<= z -1.9e+76)
       (* (* (- a (/ (* b z) j)) c) j)
       (if (<= z -4.2e-125)
         (* (fma (- z) c (* i t)) b)
         (if (<= z 5e-305)
           (* (fma (- y) j (* b t)) i)
           (if (<= z 1.9e-80)
             (* (fma (- x) a (* i b)) t)
             (if (<= z 5e+134) (* (fma (- z) x (* j i)) (- y)) t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -1.9e+177) {
		tmp = t_1;
	} else if (z <= -1.9e+76) {
		tmp = ((a - ((b * z) / j)) * c) * j;
	} else if (z <= -4.2e-125) {
		tmp = fma(-z, c, (i * t)) * b;
	} else if (z <= 5e-305) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 1.9e-80) {
		tmp = fma(-x, a, (i * b)) * t;
	} else if (z <= 5e+134) {
		tmp = fma(-z, x, (j * i)) * -y;
	} 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 <= -1.9e+177)
		tmp = t_1;
	elseif (z <= -1.9e+76)
		tmp = Float64(Float64(Float64(a - Float64(Float64(b * z) / j)) * c) * j);
	elseif (z <= -4.2e-125)
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	elseif (z <= 5e-305)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 1.9e-80)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	elseif (z <= 5e+134)
		tmp = Float64(fma(Float64(-z), x, Float64(j * i)) * Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.9e+177], t$95$1, If[LessEqual[z, -1.9e+76], N[(N[(N[(a - N[(N[(b * z), $MachinePrecision] / j), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, -4.2e-125], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 5e-305], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.9e-80], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 5e+134], N[(N[((-z) * x + N[(j * i), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -1.8999999999999999e177 or 4.99999999999999981e134 < z

   1. Initial program 61.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 79.1

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

   5. Applied rewrites 79.1%

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

   if -1.8999999999999999e177 < z < -1.90000000000000012e76

   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 j around -inf

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 79.3%

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

   if -1.90000000000000012e76 < z < -4.2e-125

   1. Initial program 82.4%

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

   3. Taylor expanded in 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 57.4

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

   5. Applied rewrites 57.4%

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

   if -4.2e-125 < z < 4.99999999999999985e-305

   1. Initial program 79.6%

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

   3. Taylor expanded in i around inf

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

      1. *-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 63.3

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

   5. Applied rewrites 63.3%

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

   if 4.99999999999999985e-305 < z < 1.89999999999999983e-80

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 60.8

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

   5. Applied rewrites 60.8%

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

   if 1.89999999999999983e-80 < z < 4.99999999999999981e134

   1. Initial program 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 61.9

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

   7. Applied rewrites 61.9%

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

Alternative 4: 50.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, j \cdot i\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -3.4e+121)
     t_1
     (if (<= z -1.3e+43)
       (* (fma (- x) t (* j c)) a)
       (if (<= z -4.2e-125)
         (* (fma (- z) c (* i t)) b)
         (if (<= z 5e-305)
           (* (fma (- y) j (* b t)) i)
           (if (<= z 1.9e-80)
             (* (fma (- x) a (* i b)) t)
             (if (<= z 5e+134) (* (fma (- z) x (* j i)) (- y)) t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -3.4e+121) {
		tmp = t_1;
	} else if (z <= -1.3e+43) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (z <= -4.2e-125) {
		tmp = fma(-z, c, (i * t)) * b;
	} else if (z <= 5e-305) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 1.9e-80) {
		tmp = fma(-x, a, (i * b)) * t;
	} else if (z <= 5e+134) {
		tmp = fma(-z, x, (j * i)) * -y;
	} 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 <= -3.4e+121)
		tmp = t_1;
	elseif (z <= -1.3e+43)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (z <= -4.2e-125)
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	elseif (z <= 5e-305)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 1.9e-80)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	elseif (z <= 5e+134)
		tmp = Float64(fma(Float64(-z), x, Float64(j * i)) * Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.4e+121], t$95$1, If[LessEqual[z, -1.3e+43], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, -4.2e-125], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 5e-305], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.9e-80], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 5e+134], N[(N[((-z) * x + N[(j * i), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -3.4000000000000001e121 or 4.99999999999999981e134 < z

   1. Initial program 62.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   if -3.4000000000000001e121 < z < -1.3000000000000001e43

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -1.3000000000000001e43 < z < -4.2e-125

   1. Initial program 85.4%

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

   3. Taylor expanded in 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 63.8

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

   5. Applied rewrites 63.8%

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

   if -4.2e-125 < z < 4.99999999999999985e-305

   1. Initial program 79.6%

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

   3. Taylor expanded in i around inf

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

      1. *-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 63.3

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

   5. Applied rewrites 63.3%

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

   if 4.99999999999999985e-305 < z < 1.89999999999999983e-80

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 60.8

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

   5. Applied rewrites 60.8%

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

   if 1.89999999999999983e-80 < z < 4.99999999999999981e134

   1. Initial program 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 61.9

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

   7. Applied rewrites 61.9%

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

Alternative 5: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{+74}:\\ \;\;\;\;\left(\left(a - \frac{b \cdot z}{j}\right) \cdot c\right) \cdot j\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -1.9e+177)
     t_1
     (if (<= z -3.8e+74)
       (* (* (- a (/ (* b z) j)) c) j)
       (if (<= z 1.02e+144)
         (fma (fma (- y) j (* b t)) i (* (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(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -1.9e+177) {
		tmp = t_1;
	} else if (z <= -3.8e+74) {
		tmp = ((a - ((b * z) / j)) * c) * j;
	} else if (z <= 1.02e+144) {
		tmp = fma(fma(-y, j, (b * t)), i, (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(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -1.9e+177)
		tmp = t_1;
	elseif (z <= -3.8e+74)
		tmp = Float64(Float64(Float64(a - Float64(Float64(b * z) / j)) * c) * j);
	elseif (z <= 1.02e+144)
		tmp = fma(fma(Float64(-y), j, Float64(b * t)), i, 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[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.9e+177], t$95$1, If[LessEqual[z, -3.8e+74], N[(N[(N[(a - N[(N[(b * z), $MachinePrecision] / j), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 1.02e+144], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8999999999999999e177 or 1.02000000000000008e144 < z

   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 81.9

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

   5. Applied rewrites 81.9%

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

   if -1.8999999999999999e177 < z < -3.7999999999999998e74

   1. Initial program 79.3%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 75.6%

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

   if -3.7999999999999998e74 < z < 1.02000000000000008e144

   1. Initial program 82.5%

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

   3. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 69.8%

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

Alternative 6: 51.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -3.4e+121)
     t_1
     (if (<= z -1.3e+43)
       (* (fma (- x) t (* j c)) a)
       (if (<= z -4.2e-125)
         (* (fma (- z) c (* i t)) b)
         (if (<= z 5e-305)
           (* (fma (- y) j (* b t)) i)
           (if (<= z 1.9e-80)
             (* (fma (- x) a (* i b)) t)
             (if (<= z 4.1e+90) (* (fma (- i) j (* z x)) y) t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -3.4e+121) {
		tmp = t_1;
	} else if (z <= -1.3e+43) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (z <= -4.2e-125) {
		tmp = fma(-z, c, (i * t)) * b;
	} else if (z <= 5e-305) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 1.9e-80) {
		tmp = fma(-x, a, (i * b)) * t;
	} else if (z <= 4.1e+90) {
		tmp = fma(-i, j, (z * x)) * y;
	} 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 <= -3.4e+121)
		tmp = t_1;
	elseif (z <= -1.3e+43)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (z <= -4.2e-125)
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	elseif (z <= 5e-305)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 1.9e-80)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	elseif (z <= 4.1e+90)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.4e+121], t$95$1, If[LessEqual[z, -1.3e+43], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, -4.2e-125], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 5e-305], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.9e-80], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 4.1e+90], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -3.4000000000000001e121 or 4.10000000000000042e90 < 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 74.3

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

   5. Applied rewrites 74.3%

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

   if -3.4000000000000001e121 < z < -1.3000000000000001e43

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -1.3000000000000001e43 < z < -4.2e-125

   1. Initial program 85.4%

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

   3. Taylor expanded in 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 63.8

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

   5. Applied rewrites 63.8%

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

   if -4.2e-125 < z < 4.99999999999999985e-305

   1. Initial program 79.6%

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

   3. Taylor expanded in i around inf

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

      1. *-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 63.3

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

   5. Applied rewrites 63.3%

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

   if 4.99999999999999985e-305 < z < 1.89999999999999983e-80

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 60.8

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

   5. Applied rewrites 60.8%

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

   if 1.89999999999999983e-80 < z < 4.10000000000000042e90

   1. Initial program 91.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   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 60.8

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

   5. Applied rewrites 60.8%

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

Alternative 7: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, j \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(j \cdot c\right) \cdot a\right)\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1450000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- i) (* j y) (fma (fma (- t) a (* z y)) x (* (* j c) a)))))
   (if (<= x -3.8e+46)
     t_1
     (if (<= x 1450000000000.0)
       (fma (fma (- z) c (* i t)) b (* (fma (- i) j (* z x)) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, (j * y), fma(fma(-t, a, (z * y)), x, ((j * c) * a)));
	double tmp;
	if (x <= -3.8e+46) {
		tmp = t_1;
	} else if (x <= 1450000000000.0) {
		tmp = fma(fma(-z, c, (i * t)), b, (fma(-i, j, (z * x)) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-i), Float64(j * y), fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(j * c) * a)))
	tmp = 0.0
	if (x <= -3.8e+46)
		tmp = t_1;
	elseif (x <= 1450000000000.0)
		tmp = fma(fma(Float64(-z), c, Float64(i * t)), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-i) * N[(j * y), $MachinePrecision] + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+46], t$95$1, If[LessEqual[x, 1450000000000.0], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999999e46 or 1.45e12 < x

   1. Initial program 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 80.3

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

   4. Applied rewrites 80.3%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 82.9%

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

   if -3.7999999999999999e46 < x < 1.45e12

   1. Initial program 75.4%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(-i, j \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(j \cdot c\right) \cdot a\right)\right)\\ \mathbf{elif}\;x \leq 1450000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(j \cdot c\right) \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot t\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- y) j (* b t)) i (* (fma (- t) a (* z y)) x))))
   (if (<= x -3.5e+46)
     t_1
     (if (<= x 5e+150)
       (fma (fma (- z) c (* i t)) b (* (fma (- i) j (* z x)) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-y, j, (b * t)), i, (fma(-t, a, (z * y)) * x));
	double tmp;
	if (x <= -3.5e+46) {
		tmp = t_1;
	} else if (x <= 5e+150) {
		tmp = fma(fma(-z, c, (i * t)), b, (fma(-i, j, (z * x)) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-y), j, Float64(b * t)), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x))
	tmp = 0.0
	if (x <= -3.5e+46)
		tmp = t_1;
	elseif (x <= 5e+150)
		tmp = fma(fma(Float64(-z), c, Float64(i * t)), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e+46], t$95$1, If[LessEqual[x, 5e+150], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999985e46 or 5.00000000000000009e150 < x

   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 c around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 82.7%

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

   if -3.49999999999999985e46 < x < 5.00000000000000009e150

   1. Initial program 76.8%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

Alternative 9: 52.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.85 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -3.4e+121)
     t_1
     (if (<= z -3.85e-62)
       (* (fma (- x) t (* j c)) a)
       (if (<= z 5e-305)
         (* (fma (- y) j (* b t)) i)
         (if (<= z 1.9e-80)
           (* (fma (- x) a (* i b)) t)
           (if (<= z 4.1e+90) (* (fma (- i) j (* z x)) y) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -3.4e+121) {
		tmp = t_1;
	} else if (z <= -3.85e-62) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (z <= 5e-305) {
		tmp = fma(-y, j, (b * t)) * i;
	} else if (z <= 1.9e-80) {
		tmp = fma(-x, a, (i * b)) * t;
	} else if (z <= 4.1e+90) {
		tmp = fma(-i, j, (z * x)) * y;
	} 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 <= -3.4e+121)
		tmp = t_1;
	elseif (z <= -3.85e-62)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (z <= 5e-305)
		tmp = Float64(fma(Float64(-y), j, Float64(b * t)) * i);
	elseif (z <= 1.9e-80)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	elseif (z <= 4.1e+90)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.4e+121], t$95$1, If[LessEqual[z, -3.85e-62], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 5e-305], N[(N[((-y) * j + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 1.9e-80], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 4.1e+90], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.4000000000000001e121 or 4.10000000000000042e90 < 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 74.3

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

   5. Applied rewrites 74.3%

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

   if -3.4000000000000001e121 < z < -3.84999999999999998e-62

   1. Initial program 82.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if -3.84999999999999998e-62 < z < 4.99999999999999985e-305

   1. Initial program 81.1%

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

   3. Taylor expanded in i around inf

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

      1. *-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 61.9

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

   5. Applied rewrites 61.9%

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

   if 4.99999999999999985e-305 < z < 1.89999999999999983e-80

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 60.8

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

   5. Applied rewrites 60.8%

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

   if 1.89999999999999983e-80 < z < 4.10000000000000042e90

   1. Initial program 91.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   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 60.8

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

   5. Applied rewrites 60.8%

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

Alternative 10: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot t, b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{j \cdot c}{t}, a \cdot x\right) \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -1.12e+126)
   (* (fma (- x) t (* j c)) a)
   (if (<= a 7.8e+14)
     (fma (* i t) b (* (fma (- i) j (* z x)) y))
     (* (fma (- a) (/ (* j c) t) (* a x)) (- t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -1.12e+126) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (a <= 7.8e+14) {
		tmp = fma((i * t), b, (fma(-i, j, (z * x)) * y));
	} else {
		tmp = fma(-a, ((j * c) / t), (a * x)) * -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -1.12e+126)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (a <= 7.8e+14)
		tmp = fma(Float64(i * t), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y));
	else
		tmp = Float64(fma(Float64(-a), Float64(Float64(j * c) / t), Float64(a * x)) * Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -1.12e+126], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 7.8e+14], N[(N[(i * t), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[((-a) * N[(N[(j * c), $MachinePrecision] / t), $MachinePrecision] + N[(a * x), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.12e126

   1. Initial program 64.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.5

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

   5. Applied rewrites 65.5%

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

   if -1.12e126 < a < 7.8e14

   1. Initial program 83.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 64.6%

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

   if 7.8e14 < a

   1. Initial program 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 69.6

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in t around -inf

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

   10. Applied rewrites 67.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot t, b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{j \cdot c}{t}, a \cdot x\right) \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -9 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-73}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -9e+113)
     t_1
     (if (<= b 5.1e-73) (+ (* (* 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 <= -9e+113) {
		tmp = t_1;
	} else if (b <= 5.1e-73) {
		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 <= -9e+113)
		tmp = t_1;
	elseif (b <= 5.1e-73)
		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, -9e+113], t$95$1, If[LessEqual[b, 5.1e-73], 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 < -9.0000000000000001e113 or 5.1e-73 < b

   1. Initial program 83.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 68.6

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

   5. Applied rewrites 68.6%

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

   if -9.0000000000000001e113 < b < 5.1e-73

   1. Initial program 73.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-73}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{if}\;a \leq -1.12 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot t, b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) t (* j c)) a)))
   (if (<= a -1.12e+126)
     t_1
     (if (<= a 7.8e+14) (fma (* i t) b (* (fma (- i) j (* z x)) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, t, (j * c)) * a;
	double tmp;
	if (a <= -1.12e+126) {
		tmp = t_1;
	} else if (a <= 7.8e+14) {
		tmp = fma((i * t), b, (fma(-i, j, (z * x)) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), t, Float64(j * c)) * a)
	tmp = 0.0
	if (a <= -1.12e+126)
		tmp = t_1;
	elseif (a <= 7.8e+14)
		tmp = fma(Float64(i * t), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.12e+126], t$95$1, If[LessEqual[a, 7.8e+14], N[(N[(i * t), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.12e126 or 7.8e14 < a

   1. Initial program 67.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if -1.12e126 < a < 7.8e14

   1. Initial program 83.0%

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

   3. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 64.6%

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

Alternative 13: 28.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot b\right) \cdot t\\ \mathbf{if}\;i \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{elif}\;i \leq 3.3 \cdot 10^{-242}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;i \leq 1.42 \cdot 10^{-89}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i b) t)))
   (if (<= i -1.7e+170)
     t_1
     (if (<= i -1.55e-27)
       (* (* (- j) y) i)
       (if (<= i 3.3e-242)
         (* (* (- z) c) b)
         (if (<= i 1.42e-89) (* (* (- x) a) 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 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -1.55e-27) {
		tmp = (-j * y) * i;
	} else if (i <= 3.3e-242) {
		tmp = (-z * c) * b;
	} else if (i <= 1.42e-89) {
		tmp = (-x * a) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * b) * t
    if (i <= (-1.7d+170)) then
        tmp = t_1
    else if (i <= (-1.55d-27)) then
        tmp = (-j * y) * i
    else if (i <= 3.3d-242) then
        tmp = (-z * c) * b
    else if (i <= 1.42d-89) then
        tmp = (-x * a) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -1.55e-27) {
		tmp = (-j * y) * i;
	} else if (i <= 3.3e-242) {
		tmp = (-z * c) * b;
	} else if (i <= 1.42e-89) {
		tmp = (-x * a) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * b) * t
	tmp = 0
	if i <= -1.7e+170:
		tmp = t_1
	elif i <= -1.55e-27:
		tmp = (-j * y) * i
	elif i <= 3.3e-242:
		tmp = (-z * c) * b
	elif i <= 1.42e-89:
		tmp = (-x * a) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * b) * t)
	tmp = 0.0
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -1.55e-27)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	elseif (i <= 3.3e-242)
		tmp = Float64(Float64(Float64(-z) * c) * b);
	elseif (i <= 1.42e-89)
		tmp = Float64(Float64(Float64(-x) * a) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * b) * t;
	tmp = 0.0;
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -1.55e-27)
		tmp = (-j * y) * i;
	elseif (i <= 3.3e-242)
		tmp = (-z * c) * b;
	elseif (i <= 1.42e-89)
		tmp = (-x * a) * t;
	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[(i * b), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[i, -1.7e+170], t$95$1, If[LessEqual[i, -1.55e-27], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[i, 3.3e-242], N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[i, 1.42e-89], N[(N[((-x) * a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.7000000000000001e170 or 1.42e-89 < i

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 49.5

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

   7. Applied rewrites 49.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.3%

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

   if -1.7000000000000001e170 < i < -1.5499999999999999e-27

   1. Initial program 72.3%

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

   3. Taylor expanded in 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 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 36.5%

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

   if -1.5499999999999999e-27 < i < 3.29999999999999982e-242

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.8%

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

   if 3.29999999999999982e-242 < i < 1.42e-89

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 48.4%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;i \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{elif}\;i \leq 3.3 \cdot 10^{-242}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;i \leq 1.42 \cdot 10^{-89}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -3.4e+121)
     t_1
     (if (<= z 1.9e-80)
       (* (fma (- x) a (* i b)) t)
       (if (<= z 4.1e+90) (* (fma (- i) j (* z x)) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -3.4e+121) {
		tmp = t_1;
	} else if (z <= 1.9e-80) {
		tmp = fma(-x, a, (i * b)) * t;
	} else if (z <= 4.1e+90) {
		tmp = fma(-i, j, (z * x)) * y;
	} 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 <= -3.4e+121)
		tmp = t_1;
	elseif (z <= 1.9e-80)
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	elseif (z <= 4.1e+90)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.4e+121], t$95$1, If[LessEqual[z, 1.9e-80], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 4.1e+90], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4000000000000001e121 or 4.10000000000000042e90 < 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 74.3

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

   5. Applied rewrites 74.3%

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

   if -3.4000000000000001e121 < z < 1.89999999999999983e-80

   1. Initial program 81.7%

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

   3. Taylor expanded in t around inf

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

      1. *-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 51.7

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

   5. Applied rewrites 51.7%

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

   if 1.89999999999999983e-80 < z < 4.10000000000000042e90

   1. Initial program 91.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   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 60.8

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

   5. Applied rewrites 60.8%

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

Alternative 15: 43.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;b \leq -5.4 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+101}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= b -5.4e+119)
     t_1
     (if (<= b 5.1e-73)
       (* (fma (- i) j (* z x)) y)
       (if (<= b 7.2e+101) (* (* b t) i) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (b <= -5.4e+119) {
		tmp = t_1;
	} else if (b <= 5.1e-73) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (b <= 7.2e+101) {
		tmp = (b * t) * i;
	} 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 (b <= -5.4e+119)
		tmp = t_1;
	elseif (b <= 5.1e-73)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (b <= 7.2e+101)
		tmp = Float64(Float64(b * t) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[b, -5.4e+119], t$95$1, If[LessEqual[b, 5.1e-73], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[b, 7.2e+101], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.3999999999999997e119 or 7.20000000000000058e101 < b

   1. Initial program 81.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   if -5.3999999999999997e119 < b < 5.1e-73

   1. Initial program 74.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

   if 5.1e-73 < b < 7.20000000000000058e101

   1. Initial program 86.4%

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

   3. Taylor expanded in 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 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.0%

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

Alternative 16: 41.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot b\right) \cdot t\\ \mathbf{if}\;i \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i b) t)))
   (if (<= i -1.7e+170)
     t_1
     (if (<= i -9.2e+91)
       (* (* (- j) y) i)
       (if (<= i 4.8e+26) (* (fma (- c) b (* y x)) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -9.2e+91) {
		tmp = (-j * y) * i;
	} else if (i <= 4.8e+26) {
		tmp = fma(-c, b, (y * x)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * b) * t)
	tmp = 0.0
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -9.2e+91)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	elseif (i <= 4.8e+26)
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[i, -1.7e+170], t$95$1, If[LessEqual[i, -9.2e+91], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[i, 4.8e+26], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.7000000000000001e170 or 4.80000000000000009e26 < i

   1. Initial program 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 73.5

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

   4. Applied rewrites 73.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 54.0

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 47.8%

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

   if -1.7000000000000001e170 < i < -9.19999999999999965e91

   1. Initial program 57.5%

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

   3. Taylor expanded in 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 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.5%

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

   if -9.19999999999999965e91 < i < 4.80000000000000009e26

   1. Initial program 81.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

Alternative 17: 29.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot b\right) \cdot t\\ \mathbf{if}\;i \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{elif}\;i \leq 6.9 \cdot 10^{-279}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i b) t)))
   (if (<= i -1.7e+170)
     t_1
     (if (<= i -1.55e-27)
       (* (* (- j) y) i)
       (if (<= i 6.9e-279)
         (* (* (- z) c) b)
         (if (<= i 8.2e-60) (* (* 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 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -1.55e-27) {
		tmp = (-j * y) * i;
	} else if (i <= 6.9e-279) {
		tmp = (-z * c) * b;
	} else if (i <= 8.2e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * b) * t
    if (i <= (-1.7d+170)) then
        tmp = t_1
    else if (i <= (-1.55d-27)) then
        tmp = (-j * y) * i
    else if (i <= 6.9d-279) then
        tmp = (-z * c) * b
    else if (i <= 8.2d-60) then
        tmp = (z * y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -1.55e-27) {
		tmp = (-j * y) * i;
	} else if (i <= 6.9e-279) {
		tmp = (-z * c) * b;
	} else if (i <= 8.2e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * b) * t
	tmp = 0
	if i <= -1.7e+170:
		tmp = t_1
	elif i <= -1.55e-27:
		tmp = (-j * y) * i
	elif i <= 6.9e-279:
		tmp = (-z * c) * b
	elif i <= 8.2e-60:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * b) * t)
	tmp = 0.0
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -1.55e-27)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	elseif (i <= 6.9e-279)
		tmp = Float64(Float64(Float64(-z) * c) * b);
	elseif (i <= 8.2e-60)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * b) * t;
	tmp = 0.0;
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -1.55e-27)
		tmp = (-j * y) * i;
	elseif (i <= 6.9e-279)
		tmp = (-z * c) * b;
	elseif (i <= 8.2e-60)
		tmp = (z * y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[i, -1.7e+170], t$95$1, If[LessEqual[i, -1.55e-27], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[i, 6.9e-279], N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[i, 8.2e-60], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.7000000000000001e170 or 8.20000000000000025e-60 < i

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.0%

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

   if -1.7000000000000001e170 < i < -1.5499999999999999e-27

   1. Initial program 72.3%

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

   3. Taylor expanded in 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 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 36.5%

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

   if -1.5499999999999999e-27 < i < 6.90000000000000024e-279

   1. Initial program 81.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.4%

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

   if 6.90000000000000024e-279 < i < 8.20000000000000025e-60

   1. Initial program 79.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   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 37.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;i \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{elif}\;i \leq 6.9 \cdot 10^{-279}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{if}\;j \leq -120000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 7 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) y (* c a)) j)))
   (if (<= j -120000000.0)
     t_1
     (if (<= j 7e-90) (* (fma (- c) b (* y x)) z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, y, (c * a)) * j;
	double tmp;
	if (j <= -120000000.0) {
		tmp = t_1;
	} else if (j <= 7e-90) {
		tmp = fma(-c, b, (y * x)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), y, Float64(c * a)) * j)
	tmp = 0.0
	if (j <= -120000000.0)
		tmp = t_1;
	elseif (j <= 7e-90)
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -120000000.0], t$95$1, If[LessEqual[j, 7e-90], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.2e8 or 6.9999999999999997e-90 < j

   1. Initial program 77.4%

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

   3. 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 32.7

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

   5. Applied rewrites 32.7%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if -1.2e8 < j < 6.9999999999999997e-90

   1. Initial program 78.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 47.1

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

   5. Applied rewrites 47.1%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -120000000:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;j \leq 7 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot b\right) \cdot t\\ \mathbf{if}\;i \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -6.7 \cdot 10^{-234}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i b) t)))
   (if (<= i -1.7e+170)
     t_1
     (if (<= i -6.7e-234)
       (* (* (- b) z) c)
       (if (<= i 8.2e-60) (* (* 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 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -6.7e-234) {
		tmp = (-b * z) * c;
	} else if (i <= 8.2e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * b) * t
    if (i <= (-1.7d+170)) then
        tmp = t_1
    else if (i <= (-6.7d-234)) then
        tmp = (-b * z) * c
    else if (i <= 8.2d-60) then
        tmp = (z * y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * b) * t;
	double tmp;
	if (i <= -1.7e+170) {
		tmp = t_1;
	} else if (i <= -6.7e-234) {
		tmp = (-b * z) * c;
	} else if (i <= 8.2e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * b) * t
	tmp = 0
	if i <= -1.7e+170:
		tmp = t_1
	elif i <= -6.7e-234:
		tmp = (-b * z) * c
	elif i <= 8.2e-60:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * b) * t)
	tmp = 0.0
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -6.7e-234)
		tmp = Float64(Float64(Float64(-b) * z) * c);
	elseif (i <= 8.2e-60)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * b) * t;
	tmp = 0.0;
	if (i <= -1.7e+170)
		tmp = t_1;
	elseif (i <= -6.7e-234)
		tmp = (-b * z) * c;
	elseif (i <= 8.2e-60)
		tmp = (z * y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[i, -1.7e+170], t$95$1, If[LessEqual[i, -6.7e-234], N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[i, 8.2e-60], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.7000000000000001e170 or 8.20000000000000025e-60 < i

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. cancel-sign-sub N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.0%

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

   if -1.7000000000000001e170 < i < -6.7e-234

   1. Initial program 79.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 43.1

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

   5. Applied rewrites 43.1%

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

   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 14.1%

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

   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 28.1%

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

   if -6.7e-234 < i < 8.20000000000000025e-60

   1. Initial program 78.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.8%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -6 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.92 \cdot 10^{+46}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 65:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -6e+149)
     t_1
     (if (<= z -1.92e+46) (* (* j c) a) (if (<= z 65.0) (* (* 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 = (z * x) * y;
	double tmp;
	if (z <= -6e+149) {
		tmp = t_1;
	} else if (z <= -1.92e+46) {
		tmp = (j * c) * a;
	} else if (z <= 65.0) {
		tmp = (i * b) * t;
	} 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 <= (-6d+149)) then
        tmp = t_1
    else if (z <= (-1.92d+46)) then
        tmp = (j * c) * a
    else if (z <= 65.0d0) then
        tmp = (i * b) * t
    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 <= -6e+149) {
		tmp = t_1;
	} else if (z <= -1.92e+46) {
		tmp = (j * c) * a;
	} else if (z <= 65.0) {
		tmp = (i * b) * t;
	} 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 <= -6e+149:
		tmp = t_1
	elif z <= -1.92e+46:
		tmp = (j * c) * a
	elif z <= 65.0:
		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(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -6e+149)
		tmp = t_1;
	elseif (z <= -1.92e+46)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 65.0)
		tmp = Float64(Float64(i * b) * t);
	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 <= -6e+149)
		tmp = t_1;
	elseif (z <= -1.92e+46)
		tmp = (j * c) * a;
	elseif (z <= 65.0)
		tmp = (i * b) * t;
	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, -6e+149], t$95$1, If[LessEqual[z, -1.92e+46], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 65.0], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000007e149 or 65 < z

   1. Initial program 70.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   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 37.5%

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

   10. Applied rewrites 39.7%

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

   if -6.00000000000000007e149 < z < -1.91999999999999992e46

   1. Initial program 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 57.2

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

   7. Applied rewrites 57.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 39.7%

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

   if -1.91999999999999992e46 < z < 65

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot i\right)\right) \cdot j + \left(c \cdot a\right) \cdot j\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y \cdot i\right), j, \left(c \cdot a\right) \cdot j\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{y \cdot i}\right), j, \left(c \cdot a\right) \cdot j\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{i \cdot y}\right), j, \left(c \cdot a\right) \cdot j\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(c \cdot a\right) \cdot j\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(c \cdot a\right) \cdot j\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\color{blue}{\left(-i\right)} \cdot y, j, \left(c \cdot a\right) \cdot j\right) \]

      12. lower-*.f64 81.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\left(-i\right) \cdot y, j, \color{blue}{\left(c \cdot a\right) \cdot j}\right) \]

   4. Applied rewrites 81.8%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\mathsf{fma}\left(\left(-i\right) \cdot y, j, \left(c \cdot a\right) \cdot j\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \color{blue}{\left(-1 \cdot b\right) \cdot i}\right) \cdot t \]

      3. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot i\right) \cdot t \]

      4. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \cdot t \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right) \cdot t} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} + b \cdot i\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) + b \cdot i\right) \cdot t \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot a} + b \cdot i\right) \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right)} \cdot a + b \cdot i\right) \cdot t \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      13. lower-*.f64 51.5

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{b \cdot i}\right) \cdot t \]

   7. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, b \cdot i\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 32.1%

       \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -2.4e-64)
   (* (* (- z) c) b)
   (if (<= z 1.2e+26) (* (* b t) i) (* (* 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 (z <= -2.4e-64) {
		tmp = (-z * c) * b;
	} else if (z <= 1.2e+26) {
		tmp = (b * t) * i;
	} 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 (z <= (-2.4d-64)) then
        tmp = (-z * c) * b
    else if (z <= 1.2d+26) then
        tmp = (b * t) * i
    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 (z <= -2.4e-64) {
		tmp = (-z * c) * b;
	} else if (z <= 1.2e+26) {
		tmp = (b * t) * i;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -2.4e-64:
		tmp = (-z * c) * b
	elif z <= 1.2e+26:
		tmp = (b * t) * i
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -2.4e-64)
		tmp = Float64(Float64(Float64(-z) * c) * b);
	elseif (z <= 1.2e+26)
		tmp = Float64(Float64(b * t) * i);
	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 (z <= -2.4e-64)
		tmp = (-z * c) * b;
	elseif (z <= 1.2e+26)
		tmp = (b * t) * i;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -2.4e-64], N[(N[((-z) * c), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 1.2e+26], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.39999999999999998e-64

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 53.7

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.2%

      \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot \color{blue}{b} \]

   if -2.39999999999999998e-64 < z < 1.20000000000000002e26

   1. Initial program 84.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-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 56.4

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot t}\right) \cdot i \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 33.1%

      \[\leadsto \left(b \cdot t\right) \cdot i \]

   if 1.20000000000000002e26 < z

   1. Initial program 68.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 69.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   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 40.5%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.8%

       \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 65:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -9.2e+78) t_1 (if (<= z 65.0) (* (* 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 = (z * x) * y;
	double tmp;
	if (z <= -9.2e+78) {
		tmp = t_1;
	} else if (z <= 65.0) {
		tmp = (i * b) * t;
	} 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 <= (-9.2d+78)) then
        tmp = t_1
    else if (z <= 65.0d0) then
        tmp = (i * b) * t
    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 <= -9.2e+78) {
		tmp = t_1;
	} else if (z <= 65.0) {
		tmp = (i * b) * t;
	} 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 <= -9.2e+78:
		tmp = t_1
	elif z <= 65.0:
		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(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -9.2e+78)
		tmp = t_1;
	elseif (z <= 65.0)
		tmp = Float64(Float64(i * b) * t);
	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 <= -9.2e+78)
		tmp = t_1;
	elseif (z <= 65.0)
		tmp = (i * b) * t;
	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, -9.2e+78], t$95$1, If[LessEqual[z, 65.0], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.2000000000000008e78 or 65 < z

   1. Initial program 70.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 66.1

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   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 35.3%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 36.4%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -9.2000000000000008e78 < z < 65

   1. Initial program 83.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{j \cdot \left(c \cdot a - y \cdot i\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \color{blue}{\left(c \cdot a - y \cdot i\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \color{blue}{\left(c \cdot a + \left(\mathsf{neg}\left(y \cdot i\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot i\right)\right) + c \cdot a\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot i\right)\right) \cdot j + \left(c \cdot a\right) \cdot j\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y \cdot i\right), j, \left(c \cdot a\right) \cdot j\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{y \cdot i}\right), j, \left(c \cdot a\right) \cdot j\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{i \cdot y}\right), j, \left(c \cdot a\right) \cdot j\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(c \cdot a\right) \cdot j\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, \left(c \cdot a\right) \cdot j\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\color{blue}{\left(-i\right)} \cdot y, j, \left(c \cdot a\right) \cdot j\right) \]

      12. lower-*.f64 81.8

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \mathsf{fma}\left(\left(-i\right) \cdot y, j, \color{blue}{\left(c \cdot a\right) \cdot j}\right) \]

   4. Applied rewrites 81.8%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{\mathsf{fma}\left(\left(-i\right) \cdot y, j, \left(c \cdot a\right) \cdot j\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   6. 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. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \color{blue}{\left(-1 \cdot b\right) \cdot i}\right) \cdot t \]

      3. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot i\right) \cdot t \]

      4. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \cdot t \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right) \cdot t} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} + b \cdot i\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) + b \cdot i\right) \cdot t \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot a} + b \cdot i\right) \cdot t \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right)} \cdot a + b \cdot i\right) \cdot t \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, b \cdot i\right)} \cdot t \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, b \cdot i\right) \cdot t \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, b \cdot i\right) \cdot t \]

      13. lower-*.f64 50.0

          \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{b \cdot i}\right) \cdot t \]

   7. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, b \cdot i\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 31.2%

       \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 30.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -1.05e+81) t_1 (if (<= z 1.2e+26) (* (* b t) i) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -1.05e+81) {
		tmp = t_1;
	} else if (z <= 1.2e+26) {
		tmp = (b * t) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-1.05d+81)) then
        tmp = t_1
    else if (z <= 1.2d+26) then
        tmp = (b * t) * i
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -1.05e+81) {
		tmp = t_1;
	} else if (z <= 1.2e+26) {
		tmp = (b * t) * i;
	} 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 <= -1.05e+81:
		tmp = t_1
	elif z <= 1.2e+26:
		tmp = (b * t) * i
	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 <= -1.05e+81)
		tmp = t_1;
	elseif (z <= 1.2e+26)
		tmp = Float64(Float64(b * t) * i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -1.05e+81)
		tmp = t_1;
	elseif (z <= 1.2e+26)
		tmp = (b * t) * i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.05e+81], t$95$1, If[LessEqual[z, 1.2e+26], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e81 or 1.20000000000000002e26 < z

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 67.6

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.6%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.7%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -1.0499999999999999e81 < z < 1.20000000000000002e26

   1. Initial program 83.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 50.6

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot t}\right) \cdot i \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(b \cdot t\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 30.1%

      \[\leadsto \left(b \cdot t\right) \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 29.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-107}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.05e+81)
   (* (* z x) y)
   (if (<= z 2.4e-107) (* (* i t) 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 <= -1.05e+81) {
		tmp = (z * x) * y;
	} else if (z <= 2.4e-107) {
		tmp = (i * t) * b;
	} 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 <= (-1.05d+81)) then
        tmp = (z * x) * y
    else if (z <= 2.4d-107) then
        tmp = (i * t) * b
    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 <= -1.05e+81) {
		tmp = (z * x) * y;
	} else if (z <= 2.4e-107) {
		tmp = (i * t) * b;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -1.05e+81:
		tmp = (z * x) * y
	elif z <= 2.4e-107:
		tmp = (i * t) * b
	else:
		tmp = (y * x) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.05e+81)
		tmp = Float64(Float64(z * x) * y);
	elseif (z <= 2.4e-107)
		tmp = Float64(Float64(i * t) * b);
	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 <= -1.05e+81)
		tmp = (z * x) * y;
	elseif (z <= 2.4e-107)
		tmp = (i * t) * b;
	else
		tmp = (y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.05e+81], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.4e-107], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0499999999999999e81

   1. Initial program 67.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 65.9

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   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 32.4%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.4%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -1.0499999999999999e81 < z < 2.39999999999999994e-107

   1. Initial program 81.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 49.9

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot t}\right) \cdot i \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.2%

      \[\leadsto \left(t \cdot i\right) \cdot \color{blue}{b} \]

   if 2.39999999999999994e-107 < z

   1. Initial program 78.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 55.6

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   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%

      \[\leadsto \left(x \cdot y\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-107}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 65:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= z -1.05e+81) t_1 (if (<= z 65.0) (* (* i t) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -1.05e+81) {
		tmp = t_1;
	} else if (z <= 65.0) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (z <= (-1.05d+81)) then
        tmp = t_1
    else if (z <= 65.0d0) then
        tmp = (i * t) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (z <= -1.05e+81) {
		tmp = t_1;
	} else if (z <= 65.0) {
		tmp = (i * t) * b;
	} 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 <= -1.05e+81:
		tmp = t_1
	elif z <= 65.0:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (z <= -1.05e+81)
		tmp = t_1;
	elseif (z <= 65.0)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (z <= -1.05e+81)
		tmp = t_1;
	elseif (z <= 65.0)
		tmp = (i * t) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.05e+81], t$95$1, If[LessEqual[z, 65.0], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e81 or 65 < z

   1. Initial program 69.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 65.7

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.6%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 36.7%

       \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -1.0499999999999999e81 < z < 65

   1. Initial program 83.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 49.3

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot t}\right) \cdot i \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot t\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.8%

      \[\leadsto \left(t \cdot i\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;z \leq 65:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 22.7% 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 77.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
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 39.9

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 39.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

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 20.0%

   \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 20.4%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
11. Add Preprocessing

Developer Target 1: 59.5% 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 2024331 
(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)))))