Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.0% → 84.1%

Time: 12.6s

Alternatives: 20

Speedup: 0.9×

• 57.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.1% accurate, 0.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $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 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 88.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 61.0%

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

4. Final simplification 84.1%

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

Alternative 2: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot a\right), b, \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\right)\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, t, i \cdot b\right), a, \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-c, \frac{z}{i}, a\right) \cdot i\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -2.55e+63)
   (fma (fma (- z) c (* i a)) b (* (fma (- a) x (* j c)) t))
   (if (<= b 2.6e+77)
     (fma
      (fma (- x) t (* i b))
      a
      (fma (fma (- i) y (* c t)) j (* (fma (- b) c (* y x)) z)))
     (* (* (fma (- c) (/ z i) a) i) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -2.55e+63) {
		tmp = fma(fma(-z, c, (i * a)), b, (fma(-a, x, (j * c)) * t));
	} else if (b <= 2.6e+77) {
		tmp = fma(fma(-x, t, (i * b)), a, fma(fma(-i, y, (c * t)), j, (fma(-b, c, (y * x)) * z)));
	} else {
		tmp = (fma(-c, (z / i), a) * i) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -2.55e+63)
		tmp = fma(fma(Float64(-z), c, Float64(i * a)), b, Float64(fma(Float64(-a), x, Float64(j * c)) * t));
	elseif (b <= 2.6e+77)
		tmp = fma(fma(Float64(-x), t, Float64(i * b)), a, fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z)));
	else
		tmp = Float64(Float64(fma(Float64(-c), Float64(z / i), a) * i) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -2.55e+63], N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e+77], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-c) * N[(z / i), $MachinePrecision] + a), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.5499999999999999e63

   1. Initial program 75.7%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 79.8%

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

   if -2.5499999999999999e63 < b < 2.6000000000000002e77

   1. Initial program 73.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 86.4%

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

   if 2.6000000000000002e77 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 79.9%

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

Alternative 3: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+121} \lor \neg \left(z \leq 6.8 \cdot 10^{-121}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -7.8e+121) (not (<= z 6.8e-121)))
   (fma (fma (- b) c (* y x)) z (* (fma (- t) x (* i b)) a))
   (fma (fma (- i) y (* c t)) j (* (fma (- a) t (* z y)) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -7.8e+121) || !(z <= 6.8e-121)) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-t, x, (i * b)) * a));
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-a, t, (z * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -7.8e+121) || !(z <= 6.8e-121))
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -7.8e+121], N[Not[LessEqual[z, 6.8e-121]], $MachinePrecision]], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.79999999999999967e121 or 6.80000000000000003e-121 < z

   1. Initial program 64.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 76.1%

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

   if -7.79999999999999967e121 < z < 6.80000000000000003e-121

   1. Initial program 83.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+121} \lor \neg \left(z \leq 6.8 \cdot 10^{-121}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;b \leq -1.06 \cdot 10^{-302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;b \leq 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot y, j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-c, \frac{z}{i}, a\right) \cdot i\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -2.9e+63)
   (* (fma i a (* (- z) c)) b)
   (if (<= b -1.06e-302)
     (fma (* y x) z (* (fma (- i) y (* c t)) j))
     (if (<= b 1e+77)
       (fma (* (- i) y) j (* (fma (- a) t (* z y)) x))
       (* (* (fma (- c) (/ z i) a) i) b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -2.9e+63) {
		tmp = fma(i, a, (-z * c)) * b;
	} else if (b <= -1.06e-302) {
		tmp = fma((y * x), z, (fma(-i, y, (c * t)) * j));
	} else if (b <= 1e+77) {
		tmp = fma((-i * y), j, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = (fma(-c, (z / i), a) * i) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -2.9e+63)
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	elseif (b <= -1.06e-302)
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	elseif (b <= 1e+77)
		tmp = fma(Float64(Float64(-i) * y), j, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = Float64(Float64(fma(Float64(-c), Float64(z / i), a) * i) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -2.9e+63], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, -1.06e-302], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+77], N[(N[((-i) * y), $MachinePrecision] * j + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-c) * N[(z / i), $MachinePrecision] + a), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.8999999999999999e63

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 60.6%

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

   if -2.8999999999999999e63 < b < -1.06e-302

   1. Initial program 68.3%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.6%

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

   if -1.06e-302 < b < 9.99999999999999983e76

   1. Initial program 80.2%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 71.2%

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

   if 9.99999999999999983e76 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 79.9%

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

Alternative 5: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -2.1e+23)
   (* (fma (- z) b (* j t)) c)
   (if (<= c 3.6e+46)
     (fma (fma (- y) j (* b a)) i (* (fma (- a) t (* z y)) x))
     (fma (fma (- i) y (* c t)) j (* (fma (- b) c (* y x)) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -2.1e+23) {
		tmp = fma(-z, b, (j * t)) * c;
	} else if (c <= 3.6e+46) {
		tmp = fma(fma(-y, j, (b * a)), i, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -2.1e+23)
		tmp = Float64(fma(Float64(-z), b, Float64(j * t)) * c);
	elseif (c <= 3.6e+46)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -2.1e+23], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 3.6e+46], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.1000000000000001e23

   1. Initial program 53.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if -2.1000000000000001e23 < c < 3.5999999999999999e46

   1. Initial program 82.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-in N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 76.7%

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

   if 3.5999999999999999e46 < c

   1. Initial program 70.1%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

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

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

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

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

      20. lower-fma.f64 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

Alternative 6: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(-i, y, c \cdot t\right)\\ \mathbf{if}\;j \leq -1.55 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, j, t\_1 \cdot z\right)\\ \mathbf{elif}\;j \leq 4.6 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- b) c (* y x))) (t_2 (fma (- i) y (* c t))))
   (if (<= j -1.55e-46)
     (fma t_2 j (* t_1 z))
     (if (<= j 4.6e+95)
       (fma t_1 z (* (fma (- t) x (* i b)) a))
       (fma t_2 j (* (fma (- a) t (* z y)) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x));
	double t_2 = fma(-i, y, (c * t));
	double tmp;
	if (j <= -1.55e-46) {
		tmp = fma(t_2, j, (t_1 * z));
	} else if (j <= 4.6e+95) {
		tmp = fma(t_1, z, (fma(-t, x, (i * b)) * a));
	} else {
		tmp = fma(t_2, j, (fma(-a, t, (z * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-b), c, Float64(y * x))
	t_2 = fma(Float64(-i), y, Float64(c * t))
	tmp = 0.0
	if (j <= -1.55e-46)
		tmp = fma(t_2, j, Float64(t_1 * z));
	elseif (j <= 4.6e+95)
		tmp = fma(t_1, z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	else
		tmp = fma(t_2, j, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.55e-46], N[(t$95$2 * j + N[(t$95$1 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 4.6e+95], N[(t$95$1 * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * j + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.55e-46

   1. Initial program 73.9%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

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

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

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

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

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

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

      20. lower-fma.f64 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.55e-46 < j < 4.59999999999999994e95

   1. Initial program 74.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 71.4%

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

   if 4.59999999999999994e95 < j

   1. Initial program 75.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.55 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{elif}\;j \leq 4.6 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{if}\;j \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, t\_1\right)\\ \mathbf{elif}\;j \leq 1.76 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) y (* c t)) j)))
   (if (<= j -1.1e-40)
     (fma (* y x) z t_1)
     (if (<= j 1.76e+205)
       (fma (fma (- b) c (* y x)) z (* (fma (- t) x (* i b)) 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, y, (c * t)) * j;
	double tmp;
	if (j <= -1.1e-40) {
		tmp = fma((y * x), z, t_1);
	} else if (j <= 1.76e+205) {
		tmp = fma(fma(-b, c, (y * x)), z, (fma(-t, x, (i * b)) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), y, Float64(c * t)) * j)
	tmp = 0.0
	if (j <= -1.1e-40)
		tmp = fma(Float64(y * x), z, t_1);
	elseif (j <= 1.76e+205)
		tmp = fma(fma(Float64(-b), c, Float64(y * x)), z, Float64(fma(Float64(-t), x, Float64(i * b)) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.1e-40], N[(N[(y * x), $MachinePrecision] * z + t$95$1), $MachinePrecision], If[LessEqual[j, 1.76e+205], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z + N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.10000000000000004e-40

   1. Initial program 73.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.1%

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

   if -1.10000000000000004e-40 < j < 1.76000000000000009e205

   1. Initial program 74.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in j around 0

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

   8. Applied rewrites 70.9%

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

   if 1.76000000000000009e205 < j

   1. Initial program 76.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{elif}\;j \leq 1.76 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-b, c, y \cdot x\right), z, \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+63} \lor \neg \left(b \leq 2.3 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -2.9e+63) (not (<= b 2.3e+70)))
   (* (fma i a (* (- z) c)) b)
   (fma (* y x) z (* (fma (- i) y (* c t)) j))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((b <= -2.9e+63) || !(b <= 2.3e+70)) {
		tmp = fma(i, a, (-z * c)) * b;
	} else {
		tmp = fma((y * x), z, (fma(-i, y, (c * t)) * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -2.9e+63) || !(b <= 2.3e+70))
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	else
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -2.9e+63], N[Not[LessEqual[b, 2.3e+70]], $MachinePrecision]], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8999999999999999e63 or 2.29999999999999994e70 < b

   1. Initial program 75.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 68.4%

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

   if -2.8999999999999999e63 < b < 2.29999999999999994e70

   1. Initial program 73.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 64.4%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+63} \lor \neg \left(b \leq 2.3 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, z, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-c, \frac{z}{i}, a\right) \cdot i\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -2.9e+63)
   (* (fma i a (* (- z) c)) b)
   (if (<= b 2.3e+70)
     (fma (* y x) z (* (fma (- i) y (* c t)) j))
     (* (* (fma (- c) (/ z i) a) i) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -2.9e+63) {
		tmp = fma(i, a, (-z * c)) * b;
	} else if (b <= 2.3e+70) {
		tmp = fma((y * x), z, (fma(-i, y, (c * t)) * j));
	} else {
		tmp = (fma(-c, (z / i), a) * i) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -2.9e+63)
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	elseif (b <= 2.3e+70)
		tmp = fma(Float64(y * x), z, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	else
		tmp = Float64(Float64(fma(Float64(-c), Float64(z / i), a) * i) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -2.9e+63], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 2.3e+70], N[(N[(y * x), $MachinePrecision] * z + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-c) * N[(z / i), $MachinePrecision] + a), $MachinePrecision] * i), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999999e63

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 60.6%

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

   if -2.8999999999999999e63 < b < 2.29999999999999994e70

   1. Initial program 73.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 64.4%

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

   if 2.29999999999999994e70 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 79.9%

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

Alternative 10: 41.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1100:\\ \;\;\;\;\mathsf{fma}\left(t, j, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;j \leq -3.25 \cdot 10^{-93} \lor \neg \left(j \leq 3.4 \cdot 10^{+29}\right):\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1100.0)
   (* (fma t j (* (- z) b)) c)
   (if (or (<= j -3.25e-93) (not (<= j 3.4e+29)))
     (* (* (- y) j) i)
     (* (fma i a (* (- z) c)) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1100.0) {
		tmp = fma(t, j, (-z * b)) * c;
	} else if ((j <= -3.25e-93) || !(j <= 3.4e+29)) {
		tmp = (-y * j) * i;
	} else {
		tmp = fma(i, a, (-z * c)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1100.0)
		tmp = Float64(fma(t, j, Float64(Float64(-z) * b)) * c);
	elseif ((j <= -3.25e-93) || !(j <= 3.4e+29))
		tmp = Float64(Float64(Float64(-y) * j) * i);
	else
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1100.0], N[(N[(t * j + N[((-z) * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[Or[LessEqual[j, -3.25e-93], N[Not[LessEqual[j, 3.4e+29]], $MachinePrecision]], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -1100

   1. Initial program 74.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 53.9

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 52.3%

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

   if -1100 < j < -3.25e-93 or 3.39999999999999981e29 < j

   1. Initial program 72.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 48.2%

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

   if -3.25e-93 < j < 3.39999999999999981e29

   1. Initial program 75.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 49.4%

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

   7. Applied rewrites 49.4%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1100:\\ \;\;\;\;\mathsf{fma}\left(t, j, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;j \leq -3.25 \cdot 10^{-93} \lor \neg \left(j \leq 3.4 \cdot 10^{+29}\right):\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-226}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma i a (* (- z) c)) b)))
   (if (<= b -1.05e-35)
     t_1
     (if (<= b 4.5e-226)
       (* (fma (- a) x (* j c)) t)
       (if (<= b 2.3e+70) (* (fma (- b) c (* y x)) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(i, a, (-z * c)) * b;
	double tmp;
	if (b <= -1.05e-35) {
		tmp = t_1;
	} else if (b <= 4.5e-226) {
		tmp = fma(-a, x, (j * c)) * t;
	} else if (b <= 2.3e+70) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(i, a, Float64(Float64(-z) * c)) * b)
	tmp = 0.0
	if (b <= -1.05e-35)
		tmp = t_1;
	elseif (b <= 4.5e-226)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	elseif (b <= 2.3e+70)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.05e-35], t$95$1, If[LessEqual[b, 4.5e-226], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 2.3e+70], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e-35 or 2.29999999999999994e70 < b

   1. Initial program 75.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 63.3%

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

   7. Applied rewrites 63.3%

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

   if -1.05e-35 < b < 4.50000000000000011e-226

   1. Initial program 68.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   if 4.50000000000000011e-226 < b < 2.29999999999999994e70

   1. Initial program 79.7%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 44.6

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

   5. Applied rewrites 44.6%

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

Alternative 12: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+77} \lor \neg \left(b \leq 2.15 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.05e+77) (not (<= b 2.15e+70)))
   (* (fma i a (* (- z) c)) b)
   (* (fma (- i) j (* z x)) y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((b <= -1.05e+77) || !(b <= 2.15e+70)) {
		tmp = fma(i, a, (-z * c)) * b;
	} else {
		tmp = fma(-i, j, (z * x)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -1.05e+77) || !(b <= 2.15e+70))
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	else
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -1.05e+77], N[Not[LessEqual[b, 2.15e+70]], $MachinePrecision]], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.0499999999999999e77 or 2.15e70 < b

   1. Initial program 76.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 69.5%

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

   7. Applied rewrites 69.5%

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

   if -1.0499999999999999e77 < b < 2.15e70

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 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. mul-1-neg N/A

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

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

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

      5. lower-fma.f64 N/A

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

      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 54.2

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

   5. Applied rewrites 54.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+77} \lor \neg \left(b \leq 2.15 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-35} \lor \neg \left(b \leq 3.1 \cdot 10^{-25}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.05e-35) (not (<= b 3.1e-25)))
   (* (fma i a (* (- z) c)) b)
   (* (fma (- a) x (* j c)) 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 ((b <= -1.05e-35) || !(b <= 3.1e-25)) {
		tmp = fma(i, a, (-z * c)) * b;
	} else {
		tmp = fma(-a, x, (j * c)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -1.05e-35) || !(b <= 3.1e-25))
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -1.05e-35], N[Not[LessEqual[b, 3.1e-25]], $MachinePrecision]], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e-35 or 3.09999999999999995e-25 < b

   1. Initial program 77.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 58.5%

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

   if -1.05e-35 < b < 3.09999999999999995e-25

   1. Initial program 71.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-35} \lor \neg \left(b \leq 3.1 \cdot 10^{-25}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-7} \lor \neg \left(b \leq 7 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -2e-7) (not (<= b 7e-9)))
   (* (fma i a (* (- z) c)) 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 ((b <= -2e-7) || !(b <= 7e-9)) {
		tmp = fma(i, a, (-z * c)) * b;
	} else {
		tmp = (-y * j) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -2e-7) || !(b <= 7e-9))
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	else
		tmp = Float64(Float64(Float64(-y) * j) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -2e-7], N[Not[LessEqual[b, 7e-9]], $MachinePrecision]], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999999e-7 or 6.9999999999999998e-9 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 58.9%

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

   if -1.9999999999999999e-7 < b < 6.9999999999999998e-9

   1. Initial program 72.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 35.0%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-7} \lor \neg \left(b \leq 7 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+77}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{+70}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -1.05e+77)
   (* (* a b) i)
   (if (<= b 2.35e+70) (* (* (- y) j) i) (* (* (- c) z) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.05e+77) {
		tmp = (a * b) * i;
	} else if (b <= 2.35e+70) {
		tmp = (-y * j) * i;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-1.05d+77)) then
        tmp = (a * b) * i
    else if (b <= 2.35d+70) then
        tmp = (-y * j) * i
    else
        tmp = (-c * z) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.05e+77) {
		tmp = (a * b) * i;
	} else if (b <= 2.35e+70) {
		tmp = (-y * j) * i;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -1.05e+77:
		tmp = (a * b) * i
	elif b <= 2.35e+70:
		tmp = (-y * j) * i
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -1.05e+77)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= 2.35e+70)
		tmp = Float64(Float64(Float64(-y) * j) * i);
	else
		tmp = Float64(Float64(Float64(-c) * z) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -1.05e+77)
		tmp = (a * b) * i;
	elseif (b <= 2.35e+70)
		tmp = (-y * j) * i;
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -1.05e+77], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 2.35e+70], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0499999999999999e77

   1. Initial program 77.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.3%

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

   if -1.0499999999999999e77 < b < 2.3499999999999999e70

   1. Initial program 73.2%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 33.1%

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

   if 2.3499999999999999e70 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.1%

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

Alternative 16: 29.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+77}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+75}:\\ \;\;\;\;\left(\left(-y\right) \cdot j\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -1.05e+77)
   (* (* a b) i)
   (if (<= b 1.7e+75) (* (* (- y) j) i) (* (* i b) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.05e+77) {
		tmp = (a * b) * i;
	} else if (b <= 1.7e+75) {
		tmp = (-y * j) * i;
	} else {
		tmp = (i * b) * a;
	}
	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 (b <= (-1.05d+77)) then
        tmp = (a * b) * i
    else if (b <= 1.7d+75) then
        tmp = (-y * j) * i
    else
        tmp = (i * b) * a
    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 (b <= -1.05e+77) {
		tmp = (a * b) * i;
	} else if (b <= 1.7e+75) {
		tmp = (-y * j) * i;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -1.05e+77:
		tmp = (a * b) * i
	elif b <= 1.7e+75:
		tmp = (-y * j) * i
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -1.05e+77)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= 1.7e+75)
		tmp = Float64(Float64(Float64(-y) * j) * i);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -1.05e+77)
		tmp = (a * b) * i;
	elseif (b <= 1.7e+75)
		tmp = (-y * j) * i;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -1.05e+77], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 1.7e+75], N[(N[((-y) * j), $MachinePrecision] * i), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0499999999999999e77

   1. Initial program 77.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.3%

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

   if -1.0499999999999999e77 < b < 1.70000000000000006e75

   1. Initial program 73.2%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 33.1%

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

   if 1.70000000000000006e75 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot a \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot a \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + b \cdot i\right) \cdot a \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot t}\right)\right) + b \cdot i\right) \cdot a \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot t} + b \cdot i\right) \cdot a \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), t, b \cdot i\right)} \cdot a \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, b \cdot i\right) \cdot a \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

      12. lower-*.f64 57.3

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 44.7%

      \[\leadsto \left(i \cdot b\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 30.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{+25} \lor \neg \left(c \leq 7.6 \cdot 10^{+15}\right):\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= c -7.2e+25) (not (<= c 7.6e+15))) (* (* j c) t) (* (* a b) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((c <= -7.2e+25) || !(c <= 7.6e+15)) {
		tmp = (j * c) * t;
	} else {
		tmp = (a * b) * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((c <= (-7.2d+25)) .or. (.not. (c <= 7.6d+15))) then
        tmp = (j * c) * t
    else
        tmp = (a * b) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((c <= -7.2e+25) || !(c <= 7.6e+15)) {
		tmp = (j * c) * t;
	} else {
		tmp = (a * b) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (c <= -7.2e+25) or not (c <= 7.6e+15):
		tmp = (j * c) * t
	else:
		tmp = (a * b) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((c <= -7.2e+25) || !(c <= 7.6e+15))
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = Float64(Float64(a * b) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((c <= -7.2e+25) || ~((c <= 7.6e+15)))
		tmp = (j * c) * t;
	else
		tmp = (a * b) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[c, -7.2e+25], N[Not[LessEqual[c, 7.6e+15]], $MachinePrecision]], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.20000000000000031e25 or 7.6e15 < c

   1. Initial program 63.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

      8. lower-*.f64 49.0

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 36.3%

      \[\leadsto \left(j \cdot c\right) \cdot t \]

   if -7.20000000000000031e25 < c < 7.6e15

   1. Initial program 82.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + \color{blue}{a \cdot b}\right) \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + a \cdot b\right) \cdot i \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + a \cdot b\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 56.1

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 29.5%

      \[\leadsto \left(a \cdot b\right) \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{+25} \lor \neg \left(c \leq 7.6 \cdot 10^{+15}\right):\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+111}:\\ \;\;\;\;\left(i \cdot a\right) \cdot b\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-115}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -9e+111)
   (* (* i a) b)
   (if (<= a 1.2e-115) (* (* j c) t) (* (* a b) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -9e+111) {
		tmp = (i * a) * b;
	} else if (a <= 1.2e-115) {
		tmp = (j * c) * t;
	} else {
		tmp = (a * b) * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-9d+111)) then
        tmp = (i * a) * b
    else if (a <= 1.2d-115) then
        tmp = (j * c) * t
    else
        tmp = (a * b) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -9e+111) {
		tmp = (i * a) * b;
	} else if (a <= 1.2e-115) {
		tmp = (j * c) * t;
	} else {
		tmp = (a * b) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -9e+111:
		tmp = (i * a) * b
	elif a <= 1.2e-115:
		tmp = (j * c) * t
	else:
		tmp = (a * b) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -9e+111)
		tmp = Float64(Float64(i * a) * b);
	elseif (a <= 1.2e-115)
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = Float64(Float64(a * b) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -9e+111)
		tmp = (i * a) * b;
	elseif (a <= 1.2e-115)
		tmp = (j * c) * t;
	else
		tmp = (a * b) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -9e+111], N[(N[(i * a), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a, 1.2e-115], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.00000000000000001e111

   1. Initial program 56.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + a \cdot i\right)} \cdot b \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)} + a \cdot i\right) \cdot b \]

      5. *-lft-identity N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(1 \cdot a\right)} \cdot i\right) \cdot b \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot a\right) \cdot i\right) \cdot b \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot a\right)\right)} \cdot i\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right) \cdot i\right) \cdot b \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right) \cdot i\right)\right)}\right) \cdot b \]

      10. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a\right)\right) \cdot i\right)\right)\right)} \cdot b \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      12. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

   5. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot i\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 52.9%

      \[\leadsto \left(i \cdot a\right) \cdot b \]

   if -9.00000000000000001e111 < a < 1.20000000000000011e-115

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

      8. lower-*.f64 33.4

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 25.3%

      \[\leadsto \left(j \cdot c\right) \cdot t \]

   if 1.20000000000000011e-115 < a

   1. Initial program 74.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + \color{blue}{a \cdot b}\right) \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + a \cdot b\right) \cdot i \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + a \cdot b\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 49.1

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 35.5%

      \[\leadsto \left(a \cdot b\right) \cdot i \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 25.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -4.5 \cdot 10^{-37}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -4.5e-37) (* (* j t) c) (* (* i b) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -4.5e-37) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	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 (j <= (-4.5d-37)) then
        tmp = (j * t) * c
    else
        tmp = (i * b) * a
    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 (j <= -4.5e-37) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if j <= -4.5e-37:
		tmp = (j * t) * c
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -4.5e-37)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (j <= -4.5e-37)
		tmp = (j * t) * c;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -4.5e-37], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -4.5000000000000004e-37

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

      8. lower-*.f64 43.5

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.1%

      \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]

   if -4.5000000000000004e-37 < j

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot a \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot a \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + b \cdot i\right) \cdot a \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot t}\right)\right) + b \cdot i\right) \cdot a \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot t} + b \cdot i\right) \cdot a \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), t, b \cdot i\right)} \cdot a \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, b \cdot i\right) \cdot a \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

      12. lower-*.f64 44.2

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

   5. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 27.2%

      \[\leadsto \left(i \cdot b\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 23.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(j \cdot t\right) \cdot c \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* j t) c))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (j * t) * c;
}

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 = (j * t) * c
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 (j * t) * c;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (j * t) * c

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(j * t) * c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (j * t) * c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 74.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

   6. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   8. lower-*.f64 36.1

      \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

5. Applied rewrites 36.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

6. Taylor expanded in x around 0

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.6%

   \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]
9. Add Preprocessing

Developer Target 1: 67.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))