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

Percentage Accurate: 74.2% → 79.4%

Time: 15.9s

Alternatives: 28

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ t_2 := \mathsf{fma}\left(-z, c, i \cdot t\right)\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, b, t\_1\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right) - \mathsf{fma}\left(-t, b, j \cdot y\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, b, \mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) j (* z x)) y)) (t_2 (fma (- z) c (* i t))))
   (if (<= b -1.15e+176)
     (fma t_2 b t_1)
     (if (<= b 2.4e-86)
       (-
        (fma (fma (- z) b (* j a)) c (* (fma (- t) a (* z y)) x))
        (* (fma (- t) b (* j y)) i))
       (fma t_2 b (fma (fma (- x) t (* j c)) a t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, j, (z * x)) * y;
	double t_2 = fma(-z, c, (i * t));
	double tmp;
	if (b <= -1.15e+176) {
		tmp = fma(t_2, b, t_1);
	} else if (b <= 2.4e-86) {
		tmp = fma(fma(-z, b, (j * a)), c, (fma(-t, a, (z * y)) * x)) - (fma(-t, b, (j * y)) * i);
	} else {
		tmp = fma(t_2, b, fma(fma(-x, t, (j * c)), a, t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), j, Float64(z * x)) * y)
	t_2 = fma(Float64(-z), c, Float64(i * t))
	tmp = 0.0
	if (b <= -1.15e+176)
		tmp = fma(t_2, b, t_1);
	elseif (b <= 2.4e-86)
		tmp = Float64(fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-t), a, Float64(z * y)) * x)) - Float64(fma(Float64(-t), b, Float64(j * y)) * i));
	else
		tmp = fma(t_2, b, fma(fma(Float64(-x), t, Float64(j * c)), a, t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.15e+176], N[(t$95$2 * b + t$95$1), $MachinePrecision], If[LessEqual[b, 2.4e-86], N[(N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(N[((-t) * b + N[(j * y), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * b + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999998e176

   1. Initial program 80.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 90.2%

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

   if -1.14999999999999998e176 < b < 2.40000000000000013e-86

   1. Initial program 77.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 86.6%

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

   if 2.40000000000000013e-86 < b

   1. Initial program 76.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.9%

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

Alternative 2: 84.5% accurate, 0.5× speedup

Math

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

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (((c * 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(fma(-j, y, (b * t)), i, (fma(-x, t, (j * c)) * a));
	}
	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 = fma(fma(Float64(-j), y, Float64(b * t)), i, Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. Taylor expanded in c around 0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.5%

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

4. Final simplification 85.9%

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

Alternative 3: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t 4.8e+151)
   (fma
    (fma (- z) c (* i t))
    b
    (fma (fma (- x) t (* j c)) a (* (fma (- i) j (* z x)) y)))
   (* (fma (- x) a (* i b)) 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 (t <= 4.8e+151) {
		tmp = fma(fma(-z, c, (i * t)), b, fma(fma(-x, t, (j * c)), a, (fma(-i, j, (z * x)) * y)));
	} else {
		tmp = fma(-x, a, (i * b)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= 4.8e+151)
		tmp = fma(fma(Float64(-z), c, Float64(i * t)), b, fma(fma(Float64(-x), t, Float64(j * c)), a, Float64(fma(Float64(-i), j, Float64(z * x)) * y)));
	else
		tmp = Float64(fma(Float64(-x), a, Float64(i * b)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, 4.8e+151], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000002e151

   1. Initial program 80.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 81.7%

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

   if 4.8000000000000002e151 < t

   1. Initial program 60.9%

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

   3. Taylor expanded in 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 89.1

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

   5. Applied rewrites 89.1%

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

Alternative 4: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(-a, \frac{j}{b}, z\right) \cdot \left(-c\right)\right) \cdot b\\ t_2 := \mathsf{fma}\left(-x, a, i \cdot b\right)\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-c\right) \cdot b\right) \cdot z\\ \mathbf{elif}\;t \leq -1.95 \cdot 10^{-165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (fma (- a) (/ j b) z) (- c)) b))
        (t_2 (fma (- x) a (* i b))))
   (if (<= t -1.05e-34)
     (fma t_2 t (* (* j c) a))
     (if (<= t -8.8e-85)
       (* (fma y x (* (- c) b)) z)
       (if (<= t -1.95e-165)
         t_1
         (if (<= t 8.2e-77)
           (* (fma (- i) j (* z x)) y)
           (if (<= t 4.1e+65) t_1 (* t_2 t))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (fma(-a, (j / b), z) * -c) * b;
	double t_2 = fma(-x, a, (i * b));
	double tmp;
	if (t <= -1.05e-34) {
		tmp = fma(t_2, t, ((j * c) * a));
	} else if (t <= -8.8e-85) {
		tmp = fma(y, x, (-c * b)) * z;
	} else if (t <= -1.95e-165) {
		tmp = t_1;
	} else if (t <= 8.2e-77) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (t <= 4.1e+65) {
		tmp = t_1;
	} else {
		tmp = t_2 * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(fma(Float64(-a), Float64(j / b), z) * Float64(-c)) * b)
	t_2 = fma(Float64(-x), a, Float64(i * b))
	tmp = 0.0
	if (t <= -1.05e-34)
		tmp = fma(t_2, t, Float64(Float64(j * c) * a));
	elseif (t <= -8.8e-85)
		tmp = Float64(fma(y, x, Float64(Float64(-c) * b)) * z);
	elseif (t <= -1.95e-165)
		tmp = t_1;
	elseif (t <= 8.2e-77)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (t <= 4.1e+65)
		tmp = t_1;
	else
		tmp = Float64(t_2 * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[((-a) * N[(j / b), $MachinePrecision] + z), $MachinePrecision] * (-c)), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.05e-34], N[(t$95$2 * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -8.8e-85], N[(N[(y * x + N[((-c) * b), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, -1.95e-165], t$95$1, If[LessEqual[t, 8.2e-77], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 4.1e+65], t$95$1, N[(t$95$2 * t), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.05e-34

   1. Initial program 74.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.4%

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

   if -1.05e-34 < t < -8.8e-85

   1. Initial program 53.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 67.7%

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

   if -8.8e-85 < t < -1.9499999999999999e-165 or 8.19999999999999925e-77 < t < 4.1000000000000001e65

   1. Initial program 84.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 82.1%

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

   9. Taylor expanded in c around -inf

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

   11. Applied rewrites 64.7%

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

   if -1.9499999999999999e-165 < t < 8.19999999999999925e-77

   1. Initial program 85.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if 4.1000000000000001e65 < t

   1. Initial program 68.2%

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

   3. Taylor expanded in 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 86.4

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

   5. Applied rewrites 86.4%

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

Alternative 5: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- z) c (* i t)) b (* (fma (- i) j (* z x)) y))))
   (if (<= b -1.7e+60)
     t_1
     (if (<= b -4.6e-285)
       (fma (fma (- z) b (* j a)) c (* (fma (- t) a (* z y)) x))
       (if (<= b 8.8e+45)
         (fma (fma (- x) a (* i b)) t (* (fma (- i) y (* c a)) j))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-z, c, (i * t)), b, (fma(-i, j, (z * x)) * y));
	double tmp;
	if (b <= -1.7e+60) {
		tmp = t_1;
	} else if (b <= -4.6e-285) {
		tmp = fma(fma(-z, b, (j * a)), c, (fma(-t, a, (z * y)) * x));
	} else if (b <= 8.8e+45) {
		tmp = fma(fma(-x, a, (i * b)), t, (fma(-i, y, (c * a)) * j));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-z), c, Float64(i * t)), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y))
	tmp = 0.0
	if (b <= -1.7e+60)
		tmp = t_1;
	elseif (b <= -4.6e-285)
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	elseif (b <= 8.8e+45)
		tmp = fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e+60], t$95$1, If[LessEqual[b, -4.6e-285], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.8e+45], N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7e60 or 8.8000000000000001e45 < b

   1. Initial program 77.7%

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

   3. Taylor expanded in 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 84.3%

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

   if -1.7e60 < b < -4.59999999999999993e-285

   1. Initial program 73.1%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. distribute-rgt-in N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 81.3%

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

   if -4.59999999999999993e-285 < b < 8.8000000000000001e45

   1. Initial program 80.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 80.4%

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

Alternative 6: 55.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ t_2 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -2.9 \cdot 10^{+208}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- x) a (* i b)) t (* (* j c) a)))
        (t_2 (* (fma (- z) c (* i t)) b)))
   (if (<= b -2.9e+208)
     t_2
     (if (<= b -3.6e-78)
       t_1
       (if (<= b 4.9e-222)
         (* (fma z y (* (- a) t)) x)
         (if (<= b 2e+89) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-x, a, (i * b)), t, ((j * c) * a));
	double t_2 = fma(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -2.9e+208) {
		tmp = t_2;
	} else if (b <= -3.6e-78) {
		tmp = t_1;
	} else if (b <= 4.9e-222) {
		tmp = fma(z, y, (-a * t)) * x;
	} else if (b <= 2e+89) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(Float64(j * c) * a))
	t_2 = Float64(fma(Float64(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -2.9e+208)
		tmp = t_2;
	elseif (b <= -3.6e-78)
		tmp = t_1;
	elseif (b <= 4.9e-222)
		tmp = Float64(fma(z, y, Float64(Float64(-a) * t)) * x);
	elseif (b <= 2e+89)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -2.9e+208], t$95$2, If[LessEqual[b, -3.6e-78], t$95$1, If[LessEqual[b, 4.9e-222], N[(N[(z * y + N[((-a) * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 2e+89], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.90000000000000008e208 or 1.99999999999999999e89 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in 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 78.0

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

   5. Applied rewrites 78.0%

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

   if -2.90000000000000008e208 < b < -3.6000000000000002e-78 or 4.9e-222 < b < 1.99999999999999999e89

   1. Initial program 80.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 59.1%

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

   if -3.6000000000000002e-78 < b < 4.9e-222

   1. Initial program 75.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 72.5

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

   8. Applied rewrites 72.5%

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

   10. Applied rewrites 72.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, i \cdot b\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot t\right), i, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq 1150:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t, \left(j \cdot c\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- x) a (* i b))))
   (if (<= t -1.8e+75)
     (fma t_1 t (* (fma (- i) y (* c a)) j))
     (if (<= t -1.2e-162)
       (fma (fma (- j) y (* b t)) i (* (fma (- x) t (* j c)) a))
       (if (<= t 1150.0)
         (+ (* (* z x) y) (* (- (* c a) (* i y)) j))
         (fma t_1 t (* (* j c) a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, a, (i * b));
	double tmp;
	if (t <= -1.8e+75) {
		tmp = fma(t_1, t, (fma(-i, y, (c * a)) * j));
	} else if (t <= -1.2e-162) {
		tmp = fma(fma(-j, y, (b * t)), i, (fma(-x, t, (j * c)) * a));
	} else if (t <= 1150.0) {
		tmp = ((z * x) * y) + (((c * a) - (i * y)) * j);
	} else {
		tmp = fma(t_1, t, ((j * c) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-x), a, Float64(i * b))
	tmp = 0.0
	if (t <= -1.8e+75)
		tmp = fma(t_1, t, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	elseif (t <= -1.2e-162)
		tmp = fma(fma(Float64(-j), y, Float64(b * t)), i, Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	elseif (t <= 1150.0)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(Float64(Float64(c * a) - Float64(i * y)) * j));
	else
		tmp = fma(t_1, t, Float64(Float64(j * c) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8e+75], N[(t$95$1 * t + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.2e-162], N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1150.0], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.8e75

   1. Initial program 75.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 90.1%

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

   if -1.8e75 < t < -1.2000000000000001e-162

   1. Initial program 70.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.5%

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

   if -1.2000000000000001e-162 < t < 1150

   1. Initial program 86.1%

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

   3. Taylor expanded in 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 68.7

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

   5. Applied rewrites 68.7%

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

   if 1150 < t

   1. Initial program 70.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot t\right), i, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq 1150:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot t\right), i, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq 1150:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- x) a (* i b)) t (* (* j c) a))))
   (if (<= t -7.4e+87)
     t_1
     (if (<= t -1.2e-162)
       (fma (fma (- j) y (* b t)) i (* (fma (- x) t (* j c)) a))
       (if (<= t 1150.0) (+ (* (* 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(fma(-x, a, (i * b)), t, ((j * c) * a));
	double tmp;
	if (t <= -7.4e+87) {
		tmp = t_1;
	} else if (t <= -1.2e-162) {
		tmp = fma(fma(-j, y, (b * t)), i, (fma(-x, t, (j * c)) * a));
	} else if (t <= 1150.0) {
		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 = fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(Float64(j * c) * a))
	tmp = 0.0
	if (t <= -7.4e+87)
		tmp = t_1;
	elseif (t <= -1.2e-162)
		tmp = fma(fma(Float64(-j), y, Float64(b * t)), i, Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	elseif (t <= 1150.0)
		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[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.4e+87], t$95$1, If[LessEqual[t, -1.2e-162], N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1150.0], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.40000000000000005e87 or 1150 < t

   1. Initial program 72.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.7%

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

   if -7.40000000000000005e87 < t < -1.2000000000000001e-162

   1. Initial program 70.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.5%

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

   if -1.2000000000000001e-162 < t < 1150

   1. Initial program 86.1%

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

   3. Taylor expanded in 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 68.7

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

   5. Applied rewrites 68.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot t\right), i, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq 1150:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot t\right), i, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;i \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.35e-18)
   (fma (fma (- j) y (* b t)) i (* (fma (- x) t (* j c)) a))
   (if (<= i 1.25e+66)
     (fma (fma (- z) b (* j a)) c (* (fma (- t) a (* z y)) x))
     (* (fma t b (* (- y) j)) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.35e-18) {
		tmp = fma(fma(-j, y, (b * t)), i, (fma(-x, t, (j * c)) * a));
	} else if (i <= 1.25e+66) {
		tmp = fma(fma(-z, b, (j * a)), c, (fma(-t, a, (z * y)) * x));
	} else {
		tmp = fma(t, b, (-y * j)) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.35e-18)
		tmp = fma(fma(Float64(-j), y, Float64(b * t)), i, Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	elseif (i <= 1.25e+66)
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	else
		tmp = Float64(fma(t, b, Float64(Float64(-y) * j)) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.35e-18], N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.25e+66], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(t * b + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.34999999999999994e-18

   1. Initial program 67.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.6%

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

   if -1.34999999999999994e-18 < i < 1.24999999999999998e66

   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 i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. distribute-rgt-in N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   if 1.24999999999999998e66 < i

   1. Initial program 74.9%

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

   3. Taylor expanded in 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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 10: 62.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1150:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- x) a (* i b)) t (* (* j c) a))))
   (if (<= t -1.75e-34)
     t_1
     (if (<= t 1150.0) (+ (* (* 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(fma(-x, a, (i * b)), t, ((j * c) * a));
	double tmp;
	if (t <= -1.75e-34) {
		tmp = t_1;
	} else if (t <= 1150.0) {
		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 = fma(fma(Float64(-x), a, Float64(i * b)), t, Float64(Float64(j * c) * a))
	tmp = 0.0
	if (t <= -1.75e-34)
		tmp = t_1;
	elseif (t <= 1150.0)
		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[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.75e-34], t$95$1, If[LessEqual[t, 1150.0], 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 t < -1.75e-34 or 1150 < t

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.8%

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

   if -1.75e-34 < t < 1150

   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 y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;t \leq 1150:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, i \cdot b\right), t, \left(j \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-179}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- b) z) c)))
   (if (<= b -3.3e+161)
     t_1
     (if (<= b -1.4e+58)
       (* (* b t) i)
       (if (<= b -1.05e-179)
         (* (* y x) z)
         (if (<= b 3.2e-43)
           (* (* (- t) x) a)
           (if (<= b 9e+123) t_1 (* (* i t) b))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-b * z) * c;
	double tmp;
	if (b <= -3.3e+161) {
		tmp = t_1;
	} else if (b <= -1.4e+58) {
		tmp = (b * t) * i;
	} else if (b <= -1.05e-179) {
		tmp = (y * x) * z;
	} else if (b <= 3.2e-43) {
		tmp = (-t * x) * a;
	} else if (b <= 9e+123) {
		tmp = t_1;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-b * z) * c
    if (b <= (-3.3d+161)) then
        tmp = t_1
    else if (b <= (-1.4d+58)) then
        tmp = (b * t) * i
    else if (b <= (-1.05d-179)) then
        tmp = (y * x) * z
    else if (b <= 3.2d-43) then
        tmp = (-t * x) * a
    else if (b <= 9d+123) then
        tmp = t_1
    else
        tmp = (i * t) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-b * z) * c;
	double tmp;
	if (b <= -3.3e+161) {
		tmp = t_1;
	} else if (b <= -1.4e+58) {
		tmp = (b * t) * i;
	} else if (b <= -1.05e-179) {
		tmp = (y * x) * z;
	} else if (b <= 3.2e-43) {
		tmp = (-t * x) * a;
	} else if (b <= 9e+123) {
		tmp = t_1;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-b * z) * c
	tmp = 0
	if b <= -3.3e+161:
		tmp = t_1
	elif b <= -1.4e+58:
		tmp = (b * t) * i
	elif b <= -1.05e-179:
		tmp = (y * x) * z
	elif b <= 3.2e-43:
		tmp = (-t * x) * a
	elif b <= 9e+123:
		tmp = t_1
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-b) * z) * c)
	tmp = 0.0
	if (b <= -3.3e+161)
		tmp = t_1;
	elseif (b <= -1.4e+58)
		tmp = Float64(Float64(b * t) * i);
	elseif (b <= -1.05e-179)
		tmp = Float64(Float64(y * x) * z);
	elseif (b <= 3.2e-43)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (b <= 9e+123)
		tmp = t_1;
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-b * z) * c;
	tmp = 0.0;
	if (b <= -3.3e+161)
		tmp = t_1;
	elseif (b <= -1.4e+58)
		tmp = (b * t) * i;
	elseif (b <= -1.05e-179)
		tmp = (y * x) * z;
	elseif (b <= 3.2e-43)
		tmp = (-t * x) * a;
	elseif (b <= 9e+123)
		tmp = t_1;
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, -3.3e+161], t$95$1, If[LessEqual[b, -1.4e+58], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, -1.05e-179], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 3.2e-43], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 9e+123], t$95$1, N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -3.29999999999999997e161 or 3.19999999999999985e-43 < b < 8.99999999999999965e123

   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 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 inf

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

   8. Applied rewrites 20.4%

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

   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 43.6%

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

   if -3.29999999999999997e161 < b < -1.3999999999999999e58

   1. Initial program 89.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.4%

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

   if -1.3999999999999999e58 < b < -1.0499999999999999e-179

   1. Initial program 69.8%

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

   3. Taylor expanded in 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 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.2%

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

   if -1.0499999999999999e-179 < b < 3.19999999999999985e-43

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.2%

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

   if 8.99999999999999965e123 < b

   1. Initial program 72.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 62.4

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

   5. Applied rewrites 62.4%

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

   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 56.1%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+161}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-179}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot t\right) \cdot b\\ t_2 := \left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-285}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(j \cdot i\right) \cdot \left(-y\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i t) b)) (t_2 (* (* (- b) z) c)))
   (if (<= b -3.1e+161)
     t_2
     (if (<= b -1.2e+85)
       t_1
       (if (<= b -3.6e-285)
         (* (* z y) x)
         (if (<= b 2.8e+70) (* (* j i) (- y)) (if (<= b 9e+123) t_2 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * t) * b;
	double t_2 = (-b * z) * c;
	double tmp;
	if (b <= -3.1e+161) {
		tmp = t_2;
	} else if (b <= -1.2e+85) {
		tmp = t_1;
	} else if (b <= -3.6e-285) {
		tmp = (z * y) * x;
	} else if (b <= 2.8e+70) {
		tmp = (j * i) * -y;
	} else if (b <= 9e+123) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (i * t) * b
    t_2 = (-b * z) * c
    if (b <= (-3.1d+161)) then
        tmp = t_2
    else if (b <= (-1.2d+85)) then
        tmp = t_1
    else if (b <= (-3.6d-285)) then
        tmp = (z * y) * x
    else if (b <= 2.8d+70) then
        tmp = (j * i) * -y
    else if (b <= 9d+123) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * t) * b;
	double t_2 = (-b * z) * c;
	double tmp;
	if (b <= -3.1e+161) {
		tmp = t_2;
	} else if (b <= -1.2e+85) {
		tmp = t_1;
	} else if (b <= -3.6e-285) {
		tmp = (z * y) * x;
	} else if (b <= 2.8e+70) {
		tmp = (j * i) * -y;
	} else if (b <= 9e+123) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * t) * b
	t_2 = (-b * z) * c
	tmp = 0
	if b <= -3.1e+161:
		tmp = t_2
	elif b <= -1.2e+85:
		tmp = t_1
	elif b <= -3.6e-285:
		tmp = (z * y) * x
	elif b <= 2.8e+70:
		tmp = (j * i) * -y
	elif b <= 9e+123:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * t) * b)
	t_2 = Float64(Float64(Float64(-b) * z) * c)
	tmp = 0.0
	if (b <= -3.1e+161)
		tmp = t_2;
	elseif (b <= -1.2e+85)
		tmp = t_1;
	elseif (b <= -3.6e-285)
		tmp = Float64(Float64(z * y) * x);
	elseif (b <= 2.8e+70)
		tmp = Float64(Float64(j * i) * Float64(-y));
	elseif (b <= 9e+123)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * t) * b;
	t_2 = (-b * z) * c;
	tmp = 0.0;
	if (b <= -3.1e+161)
		tmp = t_2;
	elseif (b <= -1.2e+85)
		tmp = t_1;
	elseif (b <= -3.6e-285)
		tmp = (z * y) * x;
	elseif (b <= 2.8e+70)
		tmp = (j * i) * -y;
	elseif (b <= 9e+123)
		tmp = t_2;
	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 * t), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, -3.1e+161], t$95$2, If[LessEqual[b, -1.2e+85], t$95$1, If[LessEqual[b, -3.6e-285], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 2.8e+70], N[(N[(j * i), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[b, 9e+123], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.10000000000000007e161 or 2.7999999999999999e70 < b < 8.99999999999999965e123

   1. Initial program 75.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 54.2

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

   5. Applied rewrites 54.2%

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

   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 19.2%

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

   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 48.7%

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

   if -3.10000000000000007e161 < b < -1.19999999999999998e85 or 8.99999999999999965e123 < b

   1. Initial program 77.7%

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

   3. Taylor expanded in 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 69.2

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

   5. Applied rewrites 69.2%

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

   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 64.0%

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

   if -1.19999999999999998e85 < b < -3.60000000000000004e-285

   1. Initial program 74.2%

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

   3. Taylor expanded in 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 40.6

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

   5. Applied rewrites 40.6%

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

   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 34.8%

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

   if -3.60000000000000004e-285 < b < 2.7999999999999999e70

   1. Initial program 81.0%

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

   3. Taylor expanded in 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 36.0

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

   5. Applied rewrites 36.0%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 34.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 30.0%

       \[\leadsto \left(i \cdot j\right) \cdot \left(-y\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+161}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+85}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-285}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;\left(j \cdot i\right) \cdot \left(-y\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-222}:\\ \;\;\;\;\left(\left(z - \frac{a \cdot t}{y}\right) \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot x, t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -1.9e+31)
     t_1
     (if (<= b 5e-222)
       (* (* (- z (/ (* a t) y)) y) x)
       (if (<= b 7.6e-96)
         (fma (* (- a) x) t (* (* j c) a))
         (if (<= b 3.1e+70) (* (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(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -1.9e+31) {
		tmp = t_1;
	} else if (b <= 5e-222) {
		tmp = ((z - ((a * t) / y)) * y) * x;
	} else if (b <= 7.6e-96) {
		tmp = fma((-a * x), t, ((j * c) * a));
	} else if (b <= 3.1e+70) {
		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(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -1.9e+31)
		tmp = t_1;
	elseif (b <= 5e-222)
		tmp = Float64(Float64(Float64(z - Float64(Float64(a * t) / y)) * y) * x);
	elseif (b <= 7.6e-96)
		tmp = fma(Float64(Float64(-a) * x), t, Float64(Float64(j * c) * a));
	elseif (b <= 3.1e+70)
		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[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.9e+31], t$95$1, If[LessEqual[b, 5e-222], N[(N[(N[(z - N[(N[(a * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 7.6e-96], N[(N[((-a) * x), $MachinePrecision] * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e+70], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.9000000000000001e31 or 3.1000000000000003e70 < b

   1. Initial program 78.5%

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

   3. Taylor expanded in 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 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.9000000000000001e31 < b < 5.00000000000000008e-222

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(y \cdot \left(z + -1 \cdot \frac{a \cdot t}{y}\right)\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 66.6%

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

   if 5.00000000000000008e-222 < b < 7.6000000000000001e-96

   1. Initial program 81.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 63.3%

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

   if 7.6000000000000001e-96 < b < 3.1000000000000003e70

   1. Initial program 83.5%

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

   3. Taylor expanded in 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 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-222}:\\ \;\;\;\;\left(\left(z - \frac{a \cdot t}{y}\right) \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot x, t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot x, t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -1.9e+31)
     t_1
     (if (<= b 5e-222)
       (* (fma z y (* (- a) t)) x)
       (if (<= b 7.6e-96)
         (fma (* (- a) x) t (* (* j c) a))
         (if (<= b 3.1e+70) (* (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(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -1.9e+31) {
		tmp = t_1;
	} else if (b <= 5e-222) {
		tmp = fma(z, y, (-a * t)) * x;
	} else if (b <= 7.6e-96) {
		tmp = fma((-a * x), t, ((j * c) * a));
	} else if (b <= 3.1e+70) {
		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(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -1.9e+31)
		tmp = t_1;
	elseif (b <= 5e-222)
		tmp = Float64(fma(z, y, Float64(Float64(-a) * t)) * x);
	elseif (b <= 7.6e-96)
		tmp = fma(Float64(Float64(-a) * x), t, Float64(Float64(j * c) * a));
	elseif (b <= 3.1e+70)
		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[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.9e+31], t$95$1, If[LessEqual[b, 5e-222], N[(N[(z * y + N[((-a) * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 7.6e-96], N[(N[((-a) * x), $MachinePrecision] * t + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e+70], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.9000000000000001e31 or 3.1000000000000003e70 < b

   1. Initial program 78.5%

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

   3. Taylor expanded in 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 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.9000000000000001e31 < b < 5.00000000000000008e-222

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   10. Applied rewrites 66.6%

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

   if 5.00000000000000008e-222 < b < 7.6000000000000001e-96

   1. Initial program 81.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 63.3%

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

   if 7.6000000000000001e-96 < b < 3.1000000000000003e70

   1. Initial program 83.5%

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

   3. Taylor expanded in 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 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\left(-a\right) \cdot x, t, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i t)) b)))
   (if (<= b -1.9e+31)
     t_1
     (if (<= b 3.4e-222)
       (* (fma z y (* (- a) t)) x)
       (if (<= b 7.6e-96)
         (* (fma (- x) t (* j c)) a)
         (if (<= b 3.1e+70) (* (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(-z, c, (i * t)) * b;
	double tmp;
	if (b <= -1.9e+31) {
		tmp = t_1;
	} else if (b <= 3.4e-222) {
		tmp = fma(z, y, (-a * t)) * x;
	} else if (b <= 7.6e-96) {
		tmp = fma(-x, t, (j * c)) * a;
	} else if (b <= 3.1e+70) {
		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(-z), c, Float64(i * t)) * b)
	tmp = 0.0
	if (b <= -1.9e+31)
		tmp = t_1;
	elseif (b <= 3.4e-222)
		tmp = Float64(fma(z, y, Float64(Float64(-a) * t)) * x);
	elseif (b <= 7.6e-96)
		tmp = Float64(fma(Float64(-x), t, Float64(j * c)) * a);
	elseif (b <= 3.1e+70)
		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[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.9e+31], t$95$1, If[LessEqual[b, 3.4e-222], N[(N[(z * y + N[((-a) * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 7.6e-96], N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 3.1e+70], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.9000000000000001e31 or 3.1000000000000003e70 < b

   1. Initial program 78.5%

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

   3. Taylor expanded in 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 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.9000000000000001e31 < b < 3.4000000000000001e-222

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   10. Applied rewrites 66.6%

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

   if 3.4000000000000001e-222 < b < 7.6000000000000001e-96

   1. Initial program 81.8%

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

   3. Taylor expanded in 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 63.0

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

   5. Applied rewrites 63.0%

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

   if 7.6000000000000001e-96 < b < 3.1000000000000003e70

   1. Initial program 83.5%

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

   3. Taylor expanded in 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 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c\\ t_2 := \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) b (* j a)) c)) (t_2 (* (fma (- x) a (* i b)) t)))
   (if (<= t -1.7e-55)
     t_2
     (if (<= t -7.5e-170)
       t_1
       (if (<= t 3.6e-77)
         (* (fma (- i) j (* z x)) y)
         (if (<= t 5e+65) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, b, (j * a)) * c;
	double t_2 = fma(-x, a, (i * b)) * t;
	double tmp;
	if (t <= -1.7e-55) {
		tmp = t_2;
	} else if (t <= -7.5e-170) {
		tmp = t_1;
	} else if (t <= 3.6e-77) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (t <= 5e+65) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), b, Float64(j * a)) * c)
	t_2 = Float64(fma(Float64(-x), a, Float64(i * b)) * t)
	tmp = 0.0
	if (t <= -1.7e-55)
		tmp = t_2;
	elseif (t <= -7.5e-170)
		tmp = t_1;
	elseif (t <= 3.6e-77)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (t <= 5e+65)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -1.7e-55], t$95$2, If[LessEqual[t, -7.5e-170], t$95$1, If[LessEqual[t, 3.6e-77], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 5e+65], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.69999999999999986e-55 or 4.99999999999999973e65 < t

   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 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 74.9

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

   5. Applied rewrites 74.9%

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

   if -1.69999999999999986e-55 < t < -7.4999999999999998e-170 or 3.6e-77 < t < 4.99999999999999973e65

   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. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if -7.4999999999999998e-170 < t < 3.6e-77

   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 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 60.4

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

   5. Applied rewrites 60.4%

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

Alternative 17: 52.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, \left(-c\right) \cdot b\right) \cdot z\\ t_2 := \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma y x (* (- c) b)) z)) (t_2 (* (fma (- x) a (* i b)) t)))
   (if (<= t -1.55e-34)
     t_2
     (if (<= t -1.3e-187)
       t_1
       (if (<= t 3e-35)
         (* (fma (- i) j (* z x)) y)
         (if (<= t 7.8e+36) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(y, x, (-c * b)) * z;
	double t_2 = fma(-x, a, (i * b)) * t;
	double tmp;
	if (t <= -1.55e-34) {
		tmp = t_2;
	} else if (t <= -1.3e-187) {
		tmp = t_1;
	} else if (t <= 3e-35) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (t <= 7.8e+36) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(y, x, Float64(Float64(-c) * b)) * z)
	t_2 = Float64(fma(Float64(-x), a, Float64(i * b)) * t)
	tmp = 0.0
	if (t <= -1.55e-34)
		tmp = t_2;
	elseif (t <= -1.3e-187)
		tmp = t_1;
	elseif (t <= 3e-35)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (t <= 7.8e+36)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(y * x + N[((-c) * b), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -1.55e-34], t$95$2, If[LessEqual[t, -1.3e-187], t$95$1, If[LessEqual[t, 3e-35], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 7.8e+36], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5499999999999999e-34 or 7.80000000000000042e36 < t

   1. Initial program 71.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -1.5499999999999999e-34 < t < -1.3e-187 or 2.99999999999999989e-35 < t < 7.80000000000000042e36

   1. Initial program 74.8%

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

   3. Taylor expanded in 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 57.5

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 57.6%

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

   if -1.3e-187 < t < 2.99999999999999989e-35

   1. Initial program 87.3%

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

   3. Taylor expanded in 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 56.5

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

   5. Applied rewrites 56.5%

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

Alternative 18: 30.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot t\right) \cdot b\\ t_2 := \left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i t) b)) (t_2 (* (* (- b) z) c)))
   (if (<= b -3.1e+161)
     t_2
     (if (<= b -1.2e+85)
       t_1
       (if (<= b 3.1e-43) (* (* z y) x) (if (<= b 9e+123) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * t) * b;
	double t_2 = (-b * z) * c;
	double tmp;
	if (b <= -3.1e+161) {
		tmp = t_2;
	} else if (b <= -1.2e+85) {
		tmp = t_1;
	} else if (b <= 3.1e-43) {
		tmp = (z * y) * x;
	} else if (b <= 9e+123) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (i * t) * b
    t_2 = (-b * z) * c
    if (b <= (-3.1d+161)) then
        tmp = t_2
    else if (b <= (-1.2d+85)) then
        tmp = t_1
    else if (b <= 3.1d-43) then
        tmp = (z * y) * x
    else if (b <= 9d+123) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * t) * b;
	double t_2 = (-b * z) * c;
	double tmp;
	if (b <= -3.1e+161) {
		tmp = t_2;
	} else if (b <= -1.2e+85) {
		tmp = t_1;
	} else if (b <= 3.1e-43) {
		tmp = (z * y) * x;
	} else if (b <= 9e+123) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * t) * b
	t_2 = (-b * z) * c
	tmp = 0
	if b <= -3.1e+161:
		tmp = t_2
	elif b <= -1.2e+85:
		tmp = t_1
	elif b <= 3.1e-43:
		tmp = (z * y) * x
	elif b <= 9e+123:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * t) * b)
	t_2 = Float64(Float64(Float64(-b) * z) * c)
	tmp = 0.0
	if (b <= -3.1e+161)
		tmp = t_2;
	elseif (b <= -1.2e+85)
		tmp = t_1;
	elseif (b <= 3.1e-43)
		tmp = Float64(Float64(z * y) * x);
	elseif (b <= 9e+123)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * t) * b;
	t_2 = (-b * z) * c;
	tmp = 0.0;
	if (b <= -3.1e+161)
		tmp = t_2;
	elseif (b <= -1.2e+85)
		tmp = t_1;
	elseif (b <= 3.1e-43)
		tmp = (z * y) * x;
	elseif (b <= 9e+123)
		tmp = t_2;
	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 * t), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, -3.1e+161], t$95$2, If[LessEqual[b, -1.2e+85], t$95$1, If[LessEqual[b, 3.1e-43], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 9e+123], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.10000000000000007e161 or 3.0999999999999999e-43 < b < 8.99999999999999965e123

   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 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 inf

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

   8. Applied rewrites 20.4%

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

   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 43.6%

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

   if -3.10000000000000007e161 < b < -1.19999999999999998e85 or 8.99999999999999965e123 < b

   1. Initial program 77.7%

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

   3. Taylor expanded in 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 69.2

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

   5. Applied rewrites 69.2%

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

   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 64.0%

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

   if -1.19999999999999998e85 < b < 3.0999999999999999e-43

   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 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 32.8

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

   5. Applied rewrites 32.8%

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

   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 28.9%

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

Alternative 19: 30.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+71}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- c) z) b)))
   (if (<= b -6.5e+162)
     t_1
     (if (<= b -1.4e+58)
       (* (* b t) i)
       (if (<= b 3.4e+71)
         (* (* y x) z)
         (if (<= b 9e+123) t_1 (* (* i t) b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-c * z) * b;
	double tmp;
	if (b <= -6.5e+162) {
		tmp = t_1;
	} else if (b <= -1.4e+58) {
		tmp = (b * t) * i;
	} else if (b <= 3.4e+71) {
		tmp = (y * x) * z;
	} else if (b <= 9e+123) {
		tmp = t_1;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-c * z) * b
    if (b <= (-6.5d+162)) then
        tmp = t_1
    else if (b <= (-1.4d+58)) then
        tmp = (b * t) * i
    else if (b <= 3.4d+71) then
        tmp = (y * x) * z
    else if (b <= 9d+123) then
        tmp = t_1
    else
        tmp = (i * t) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-c * z) * b;
	double tmp;
	if (b <= -6.5e+162) {
		tmp = t_1;
	} else if (b <= -1.4e+58) {
		tmp = (b * t) * i;
	} else if (b <= 3.4e+71) {
		tmp = (y * x) * z;
	} else if (b <= 9e+123) {
		tmp = t_1;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-c * z) * b
	tmp = 0
	if b <= -6.5e+162:
		tmp = t_1
	elif b <= -1.4e+58:
		tmp = (b * t) * i
	elif b <= 3.4e+71:
		tmp = (y * x) * z
	elif b <= 9e+123:
		tmp = t_1
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-c) * z) * b)
	tmp = 0.0
	if (b <= -6.5e+162)
		tmp = t_1;
	elseif (b <= -1.4e+58)
		tmp = Float64(Float64(b * t) * i);
	elseif (b <= 3.4e+71)
		tmp = Float64(Float64(y * x) * z);
	elseif (b <= 9e+123)
		tmp = t_1;
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-c * z) * b;
	tmp = 0.0;
	if (b <= -6.5e+162)
		tmp = t_1;
	elseif (b <= -1.4e+58)
		tmp = (b * t) * i;
	elseif (b <= 3.4e+71)
		tmp = (y * x) * z;
	elseif (b <= 9e+123)
		tmp = t_1;
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -6.5e+162], t$95$1, If[LessEqual[b, -1.4e+58], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 3.4e+71], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 9e+123], t$95$1, N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000004e162 or 3.3999999999999998e71 < b < 8.99999999999999965e123

   1. Initial program 75.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 54.2

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

   5. Applied rewrites 54.2%

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

   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 47.9%

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

   if -6.5000000000000004e162 < b < -1.3999999999999999e58

   1. Initial program 89.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.4%

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

   if -1.3999999999999999e58 < b < 3.3999999999999998e71

   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 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 28.2%

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

   if 8.99999999999999965e123 < b

   1. Initial program 72.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 62.4

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

   5. Applied rewrites 62.4%

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

   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 56.1%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+162}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+71}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+123}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 49.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{if}\;i \leq -6 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.8 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;i \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-c\right) \cdot b\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma t b (* (- y) j)) i)))
   (if (<= i -6e-129)
     t_1
     (if (<= i -1.8e-201)
       (* (* (- t) x) a)
       (if (<= i 1.3e+33) (* (fma y x (* (- c) b)) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(t, b, (-y * j)) * i;
	double tmp;
	if (i <= -6e-129) {
		tmp = t_1;
	} else if (i <= -1.8e-201) {
		tmp = (-t * x) * a;
	} else if (i <= 1.3e+33) {
		tmp = fma(y, x, (-c * b)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(t, b, Float64(Float64(-y) * j)) * i)
	tmp = 0.0
	if (i <= -6e-129)
		tmp = t_1;
	elseif (i <= -1.8e-201)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (i <= 1.3e+33)
		tmp = Float64(fma(y, x, Float64(Float64(-c) * b)) * 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[(t * b + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -6e-129], t$95$1, If[LessEqual[i, -1.8e-201], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[i, 1.3e+33], N[(N[(y * x + N[((-c) * b), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.9999999999999996e-129 or 1.2999999999999999e33 < i

   1. Initial program 71.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 57.6%

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

   if -5.9999999999999996e-129 < i < -1.80000000000000016e-201

   1. Initial program 77.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. +-commutative N/A

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.8%

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

   if -1.80000000000000016e-201 < i < 1.2999999999999999e33

   1. Initial program 86.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 48.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{elif}\;i \leq -1.8 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;i \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-c\right) \cdot b\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{if}\;i \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma t b (* (- y) j)) i)))
   (if (<= i -2.6e-8)
     t_1
     (if (<= i 2.1e+63) (* (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(t, b, (-y * j)) * i;
	double tmp;
	if (i <= -2.6e-8) {
		tmp = t_1;
	} else if (i <= 2.1e+63) {
		tmp = 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(t, b, Float64(Float64(-y) * j)) * i)
	tmp = 0.0
	if (i <= -2.6e-8)
		tmp = t_1;
	elseif (i <= 2.1e+63)
		tmp = 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[(t * b + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -2.6e-8], t$95$1, If[LessEqual[i, 2.1e+63], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.6000000000000001e-8 or 2.1000000000000002e63 < i

   1. Initial program 70.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 65.6%

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

   if -2.6000000000000001e-8 < i < 2.1000000000000002e63

   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 y around 0

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 54.8

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

   8. Applied rewrites 54.8%

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

Alternative 22: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{if}\;i \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma t b (* (- y) j)) i)))
   (if (<= i -2.6e-8)
     t_1
     (if (<= i 2.8e+62) (* (fma z y (* (- a) t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(t, b, (-y * j)) * i;
	double tmp;
	if (i <= -2.6e-8) {
		tmp = t_1;
	} else if (i <= 2.8e+62) {
		tmp = fma(z, y, (-a * t)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(t, b, Float64(Float64(-y) * j)) * i)
	tmp = 0.0
	if (i <= -2.6e-8)
		tmp = t_1;
	elseif (i <= 2.8e+62)
		tmp = Float64(fma(z, y, Float64(Float64(-a) * t)) * 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[(t * b + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -2.6e-8], t$95$1, If[LessEqual[i, 2.8e+62], N[(N[(z * y + N[((-a) * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.6000000000000001e-8 or 2.80000000000000014e62 < i

   1. Initial program 70.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 65.6%

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

   if -2.6000000000000001e-8 < i < 2.80000000000000014e62

   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 y around 0

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 54.8

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

   8. Applied rewrites 54.8%

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

   10. Applied rewrites 54.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \left(-a\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, b, \left(-y\right) \cdot j\right) \cdot i\\ \mathbf{if}\;i \leq -6.5 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma t b (* (- y) j)) i)))
   (if (<= i -6.5e-129) t_1 (if (<= i 1.9e-10) (* (* (- a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(t, b, (-y * j)) * i;
	double tmp;
	if (i <= -6.5e-129) {
		tmp = t_1;
	} else if (i <= 1.9e-10) {
		tmp = (-a * t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(t, b, Float64(Float64(-y) * j)) * i)
	tmp = 0.0
	if (i <= -6.5e-129)
		tmp = t_1;
	elseif (i <= 1.9e-10)
		tmp = Float64(Float64(Float64(-a) * t) * 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[(t * b + N[((-y) * j), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -6.5e-129], t$95$1, If[LessEqual[i, 1.9e-10], N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -6.49999999999999952e-129 or 1.8999999999999999e-10 < i

   1. Initial program 73.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.4%

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

   if -6.49999999999999952e-129 < i < 1.8999999999999999e-10

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 39.8%

       \[\leadsto \left(\left(-a\right) \cdot t\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 31.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-34}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;t \leq 4200000000:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot t\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -1.66e-34)
   (* (* i t) b)
   (if (<= t 4200000000.0) (* (* z y) x) (* (* b t) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -1.66e-34) {
		tmp = (i * t) * b;
	} else if (t <= 4200000000.0) {
		tmp = (z * y) * x;
	} else {
		tmp = (b * t) * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (t <= (-1.66d-34)) then
        tmp = (i * t) * b
    else if (t <= 4200000000.0d0) then
        tmp = (z * y) * x
    else
        tmp = (b * t) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -1.66e-34) {
		tmp = (i * t) * b;
	} else if (t <= 4200000000.0) {
		tmp = (z * y) * x;
	} else {
		tmp = (b * t) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -1.66e-34:
		tmp = (i * t) * b
	elif t <= 4200000000.0:
		tmp = (z * y) * x
	else:
		tmp = (b * t) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -1.66e-34)
		tmp = Float64(Float64(i * t) * b);
	elseif (t <= 4200000000.0)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(b * t) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= -1.66e-34)
		tmp = (i * t) * b;
	elseif (t <= 4200000000.0)
		tmp = (z * y) * x;
	else
		tmp = (b * t) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -1.66e-34], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t, 4200000000.0], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6599999999999999e-34

   1. Initial program 74.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 46.0

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

   5. Applied rewrites 46.0%

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

   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 39.4%

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

   if -1.6599999999999999e-34 < t < 4.2e9

   1. Initial program 82.1%

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

   3. Taylor expanded in 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 49.1

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

   5. Applied rewrites 49.1%

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

   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 31.0%

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

   if 4.2e9 < t

   1. Initial program 69.7%

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

   3. Taylor expanded in 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 40.1

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.8%

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

Alternative 25: 29.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+70}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i t) b)))
   (if (<= b -1.4e+58) t_1 (if (<= b 7.8e+70) (* (* 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 * t) * b;
	double tmp;
	if (b <= -1.4e+58) {
		tmp = t_1;
	} else if (b <= 7.8e+70) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * t) * b
    if (b <= (-1.4d+58)) then
        tmp = t_1
    else if (b <= 7.8d+70) then
        tmp = (y * x) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * t) * b;
	double tmp;
	if (b <= -1.4e+58) {
		tmp = t_1;
	} else if (b <= 7.8e+70) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * t) * b
	tmp = 0
	if b <= -1.4e+58:
		tmp = t_1
	elif b <= 7.8e+70:
		tmp = (y * x) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * t) * b)
	tmp = 0.0
	if (b <= -1.4e+58)
		tmp = t_1;
	elseif (b <= 7.8e+70)
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * t) * b;
	tmp = 0.0;
	if (b <= -1.4e+58)
		tmp = t_1;
	elseif (b <= 7.8e+70)
		tmp = (y * x) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.4e+58], t$95$1, If[LessEqual[b, 7.8e+70], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999e58 or 7.79999999999999949e70 < b

   1. Initial program 77.1%

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

   3. Taylor expanded in 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 53.4

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

   5. Applied rewrites 53.4%

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

   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 42.0%

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

   if -1.3999999999999999e58 < b < 7.79999999999999949e70

   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 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 28.2%

      \[\leadsto \left(y \cdot x\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 29.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot t\right) \cdot b\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+70}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i t) b)))
   (if (<= b -1.2e+85) t_1 (if (<= b 3e+70) (* (* 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 * t) * b;
	double tmp;
	if (b <= -1.2e+85) {
		tmp = t_1;
	} else if (b <= 3e+70) {
		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 * t) * b
    if (b <= (-1.2d+85)) then
        tmp = t_1
    else if (b <= 3d+70) 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 * t) * b;
	double tmp;
	if (b <= -1.2e+85) {
		tmp = t_1;
	} else if (b <= 3e+70) {
		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 * t) * b
	tmp = 0
	if b <= -1.2e+85:
		tmp = t_1
	elif b <= 3e+70:
		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 * t) * b)
	tmp = 0.0
	if (b <= -1.2e+85)
		tmp = t_1;
	elseif (b <= 3e+70)
		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 * t) * b;
	tmp = 0.0;
	if (b <= -1.2e+85)
		tmp = t_1;
	elseif (b <= 3e+70)
		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 * t), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.2e+85], t$95$1, If[LessEqual[b, 3e+70], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.19999999999999998e85 or 2.99999999999999976e70 < b

   1. Initial program 76.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 53.0

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

   5. Applied rewrites 53.0%

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

   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 42.2%

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

   if -1.19999999999999998e85 < b < 2.99999999999999976e70

   1. Initial program 77.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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 34.8

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

   5. Applied rewrites 34.8%

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

   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 28.3%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 23.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 77.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
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 38.4

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

5. Applied rewrites 38.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 22.9%

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

Alternative 28: 22.6% 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 77.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
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 38.4

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

5. Applied rewrites 38.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 22.9%

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

10. Applied rewrites 20.9%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
11. Add Preprocessing

Developer Target 1: 59.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024294 
(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)))))