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

Percentage Accurate: 73.4% → 84.4%

Time: 14.5s

Alternatives: 19

Speedup: 0.9×

• 57.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.0%

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

Alternative 2: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.08 \cdot 10^{+256}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \leq -1.95 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-y, i, c \cdot a\right), \frac{j}{x}, z \cdot y\right)\right) \cdot x\\ \mathbf{elif}\;j \leq 0.19:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1.08e+256)
   (+ (* (* i t) b) (* j (- (* c a) (* y i))))
   (if (<= j -1.95e+21)
     (* (fma (- a) t (fma (fma (- y) i (* c a)) (/ j x) (* z y))) x)
     (if (<= j 0.19)
       (+
        (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
        (* (* j c) a))
       (fma (fma (- t) a (* z y)) x (* (fma (- i) y (* c a)) 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 (j <= -1.08e+256) {
		tmp = ((i * t) * b) + (j * ((c * a) - (y * i)));
	} else if (j <= -1.95e+21) {
		tmp = fma(-a, t, fma(fma(-y, i, (c * a)), (j / x), (z * y))) * x;
	} else if (j <= 0.19) {
		tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + ((j * c) * a);
	} else {
		tmp = fma(fma(-t, a, (z * y)), x, (fma(-i, y, (c * a)) * j));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1.08e+256)
		tmp = Float64(Float64(Float64(i * t) * b) + Float64(j * Float64(Float64(c * a) - Float64(y * i))));
	elseif (j <= -1.95e+21)
		tmp = Float64(fma(Float64(-a), t, fma(fma(Float64(-y), i, Float64(c * a)), Float64(j / x), Float64(z * y))) * x);
	elseif (j <= 0.19)
		tmp = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(Float64(j * c) * a));
	else
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1.08e+256], N[(N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, -1.95e+21], N[(N[((-a) * t + N[(N[((-y) * i + N[(c * a), $MachinePrecision]), $MachinePrecision] * N[(j / x), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[j, 0.19], N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.08e256

   1. Initial program 82.1%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -1.08e256 < j < -1.95e21

   1. Initial program 79.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 83.5%

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

   if -1.95e21 < j < 0.19

   1. Initial program 75.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if 0.19 < j

   1. Initial program 76.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

Alternative 3: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(\mathsf{fma}\left(-x, t, j \cdot c\right), a, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-b, z, j \cdot a\right), \frac{c}{x}, z \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -2e+146)
   (fma (fma (- z) b (* j a)) c (* (fma (- t) a (* z y)) x))
   (if (<= x 1.12e+61)
     (fma
      (fma (- z) c (* i t))
      b
      (fma (fma (- x) t (* j c)) a (* (fma (- i) j (* z x)) y)))
     (* (fma (- a) t (fma (fma (- b) z (* j a)) (/ c x) (* 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 (x <= -2e+146) {
		tmp = fma(fma(-z, b, (j * a)), c, (fma(-t, a, (z * y)) * x));
	} else if (x <= 1.12e+61) {
		tmp = fma(fma(-z, c, (i * t)), b, fma(fma(-x, t, (j * c)), a, (fma(-i, j, (z * x)) * y)));
	} else {
		tmp = fma(-a, t, fma(fma(-b, z, (j * a)), (c / x), (z * y))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -2e+146)
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	elseif (x <= 1.12e+61)
		tmp = fma(fma(Float64(-z), c, Float64(i * t)), b, fma(fma(Float64(-x), t, Float64(j * c)), a, Float64(fma(Float64(-i), j, Float64(z * x)) * y)));
	else
		tmp = Float64(fma(Float64(-a), t, fma(fma(Float64(-b), z, Float64(j * a)), Float64(c / x), Float64(z * y))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -2e+146], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e+61], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) * t + N[(N[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * N[(c / x), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999987e146

   1. Initial program 77.7%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 87.9%

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

   if -1.99999999999999987e146 < x < 1.12e61

   1. Initial program 79.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 85.5%

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

   if 1.12e61 < x

   1. Initial program 68.3%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.5%

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

Alternative 4: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right)\\ t_2 := \mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -1.32 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, t\_1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- t) a (* z y))) (t_2 (* (fma t i (* (- c) z)) b)))
   (if (<= b -1.32e+122)
     t_2
     (if (<= b 7e-169)
       (fma t_1 x (* (fma (- i) y (* c a)) j))
       (if (<= b 4e+113) (fma (fma (- z) b (* j a)) c (* t_1 x)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y));
	double t_2 = fma(t, i, (-c * z)) * b;
	double tmp;
	if (b <= -1.32e+122) {
		tmp = t_2;
	} else if (b <= 7e-169) {
		tmp = fma(t_1, x, (fma(-i, y, (c * a)) * j));
	} else if (b <= 4e+113) {
		tmp = fma(fma(-z, b, (j * a)), c, (t_1 * x));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-t), a, Float64(z * y))
	t_2 = Float64(fma(t, i, Float64(Float64(-c) * z)) * b)
	tmp = 0.0
	if (b <= -1.32e+122)
		tmp = t_2;
	elseif (b <= 7e-169)
		tmp = fma(t_1, x, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	elseif (b <= 4e+113)
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(t_1 * x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * i + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.32e+122], t$95$2, If[LessEqual[b, 7e-169], N[(t$95$1 * x + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+113], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(t$95$1 * x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.31999999999999992e122 or 4e113 < b

   1. Initial program 79.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.7%

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

   if -1.31999999999999992e122 < b < 7.0000000000000006e-169

   1. Initial program 76.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 7.0000000000000006e-169 < b < 4e113

   1. Initial program 73.8%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 72.5%

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

Alternative 5: 61.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-z\right) \cdot b\right) \cdot c + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{if}\;c \leq -4 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(\left(-i\right) \cdot j\right) \cdot y\right)\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \left(\left(-a\right) \cdot t\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (+ (* (* (- z) b) c) (* j (- (* c a) (* y i))))))
   (if (<= c -4e-32)
     t_1
     (if (<= c 5.3e-11)
       (fma (fma (- t) a (* z y)) x (* (* (- i) j) y))
       (if (<= c 2.9e+181)
         t_1
         (fma (fma (- z) b (* j a)) c (* (* (- a) t) 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 = ((-z * b) * c) + (j * ((c * a) - (y * i)));
	double tmp;
	if (c <= -4e-32) {
		tmp = t_1;
	} else if (c <= 5.3e-11) {
		tmp = fma(fma(-t, a, (z * y)), x, ((-i * j) * y));
	} else if (c <= 2.9e+181) {
		tmp = t_1;
	} else {
		tmp = fma(fma(-z, b, (j * a)), c, ((-a * t) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(Float64(-z) * b) * c) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
	tmp = 0.0
	if (c <= -4e-32)
		tmp = t_1;
	elseif (c <= 5.3e-11)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(Float64(-i) * j) * y));
	elseif (c <= 2.9e+181)
		tmp = t_1;
	else
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(Float64(Float64(-a) * t) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[((-z) * b), $MachinePrecision] * c), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4e-32], t$95$1, If[LessEqual[c, 5.3e-11], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.9e+181], t$95$1, N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.00000000000000022e-32 or 5.2999999999999998e-11 < c < 2.9e181

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -4.00000000000000022e-32 < c < 5.2999999999999998e-11

   1. Initial program 81.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.1%

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

   if 2.9e181 < c

   1. Initial program 62.9%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 87.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(\left(-i\right) \cdot j\right) \cdot y\right)\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{+181}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \left(\left(-a\right) \cdot t\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 3.35 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -1.32e+122)
   (* (fma t i (* (- c) z)) b)
   (if (<= b 3.35e+33)
     (fma (fma (- t) a (* z y)) x (* (fma (- i) y (* c a)) j))
     (* (fma (- z) c (* i t)) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.32e+122) {
		tmp = fma(t, i, (-c * z)) * b;
	} else if (b <= 3.35e+33) {
		tmp = fma(fma(-t, a, (z * y)), x, (fma(-i, y, (c * a)) * j));
	} else {
		tmp = fma(-z, c, (i * t)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -1.32e+122)
		tmp = Float64(fma(t, i, Float64(Float64(-c) * z)) * b);
	elseif (b <= 3.35e+33)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-i), y, Float64(c * a)) * j));
	else
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -1.32e+122], N[(N[(t * i + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 3.35e+33], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.31999999999999992e122

   1. Initial program 85.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 85.6%

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

   if -1.31999999999999992e122 < b < 3.35e33

   1. Initial program 75.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if 3.35e33 < b

   1. Initial program 74.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

Alternative 7: 59.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 3.35 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(\left(-i\right) \cdot j\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -2.3e+138)
   (* (fma t i (* (- c) z)) b)
   (if (<= b 3.35e+33)
     (fma (fma (- t) a (* z y)) x (* (* (- i) j) y))
     (* (fma (- z) c (* i t)) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -2.3e+138) {
		tmp = fma(t, i, (-c * z)) * b;
	} else if (b <= 3.35e+33) {
		tmp = fma(fma(-t, a, (z * y)), x, ((-i * j) * y));
	} else {
		tmp = fma(-z, c, (i * t)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -2.3e+138)
		tmp = Float64(fma(t, i, Float64(Float64(-c) * z)) * b);
	elseif (b <= 3.35e+33)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(Float64(-i) * j) * y));
	else
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -2.3e+138], N[(N[(t * i + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 3.35e+33], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.30000000000000008e138

   1. Initial program 84.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.1%

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

   if -2.30000000000000008e138 < b < 3.35e33

   1. Initial program 75.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

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

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

      17. +-commutative N/A

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

      18. neg-mul-1 N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.5%

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

   if 3.35e33 < b

   1. Initial program 74.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 3.35 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(\left(-i\right) \cdot j\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot i\right) \cdot b\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-214}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-95}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{elif}\;b \leq 0.152:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* t i) b)))
   (if (<= b -3.8e+122)
     t_1
     (if (<= b -4.2e-214)
       (* (* (- y) i) j)
       (if (<= b 2.85e-95)
         (* (* x y) z)
         (if (<= b 0.152)
           (* (* (- i) j) y)
           (if (<= b 2.9e+187) (* (* (- b) c) 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 = (t * i) * b;
	double tmp;
	if (b <= -3.8e+122) {
		tmp = t_1;
	} else if (b <= -4.2e-214) {
		tmp = (-y * i) * j;
	} else if (b <= 2.85e-95) {
		tmp = (x * y) * z;
	} else if (b <= 0.152) {
		tmp = (-i * j) * y;
	} else if (b <= 2.9e+187) {
		tmp = (-b * c) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * i) * b
    if (b <= (-3.8d+122)) then
        tmp = t_1
    else if (b <= (-4.2d-214)) then
        tmp = (-y * i) * j
    else if (b <= 2.85d-95) then
        tmp = (x * y) * z
    else if (b <= 0.152d0) then
        tmp = (-i * j) * y
    else if (b <= 2.9d+187) then
        tmp = (-b * c) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (t * i) * b;
	double tmp;
	if (b <= -3.8e+122) {
		tmp = t_1;
	} else if (b <= -4.2e-214) {
		tmp = (-y * i) * j;
	} else if (b <= 2.85e-95) {
		tmp = (x * y) * z;
	} else if (b <= 0.152) {
		tmp = (-i * j) * y;
	} else if (b <= 2.9e+187) {
		tmp = (-b * c) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (t * i) * b
	tmp = 0
	if b <= -3.8e+122:
		tmp = t_1
	elif b <= -4.2e-214:
		tmp = (-y * i) * j
	elif b <= 2.85e-95:
		tmp = (x * y) * z
	elif b <= 0.152:
		tmp = (-i * j) * y
	elif b <= 2.9e+187:
		tmp = (-b * c) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(t * i) * b)
	tmp = 0.0
	if (b <= -3.8e+122)
		tmp = t_1;
	elseif (b <= -4.2e-214)
		tmp = Float64(Float64(Float64(-y) * i) * j);
	elseif (b <= 2.85e-95)
		tmp = Float64(Float64(x * y) * z);
	elseif (b <= 0.152)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (b <= 2.9e+187)
		tmp = Float64(Float64(Float64(-b) * c) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (t * i) * b;
	tmp = 0.0;
	if (b <= -3.8e+122)
		tmp = t_1;
	elseif (b <= -4.2e-214)
		tmp = (-y * i) * j;
	elseif (b <= 2.85e-95)
		tmp = (x * y) * z;
	elseif (b <= 0.152)
		tmp = (-i * j) * y;
	elseif (b <= 2.9e+187)
		tmp = (-b * c) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(t * i), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -3.8e+122], t$95$1, If[LessEqual[b, -4.2e-214], N[(N[((-y) * i), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[b, 2.85e-95], N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 0.152], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[b, 2.9e+187], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -3.7999999999999998e122 or 2.9000000000000001e187 < b

   1. Initial program 79.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 52.9%

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

   if -3.7999999999999998e122 < b < -4.19999999999999984e-214

   1. Initial program 73.1%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 48.9

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

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 38.1%

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

   if -4.19999999999999984e-214 < b < 2.85e-95

   1. Initial program 74.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 41.6%

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

   if 2.85e-95 < b < 0.151999999999999996

   1. Initial program 86.0%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in y around inf

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

      1. *-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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.3

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.5%

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

   if 0.151999999999999996 < b < 2.9000000000000001e187

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.6%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 28.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-y\right) \cdot i\right) \cdot j\\ t_2 := \left(t \cdot i\right) \cdot b\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-95}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{elif}\;b \leq 0.245:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- y) i) j)) (t_2 (* (* t i) b)))
   (if (<= b -3.8e+122)
     t_2
     (if (<= b -4.2e-214)
       t_1
       (if (<= b 9.5e-95)
         (* (* x y) z)
         (if (<= b 0.245) t_1 (if (<= b 2.9e+187) (* (* (- b) c) z) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-y * i) * j;
	double t_2 = (t * i) * b;
	double tmp;
	if (b <= -3.8e+122) {
		tmp = t_2;
	} else if (b <= -4.2e-214) {
		tmp = t_1;
	} else if (b <= 9.5e-95) {
		tmp = (x * y) * z;
	} else if (b <= 0.245) {
		tmp = t_1;
	} else if (b <= 2.9e+187) {
		tmp = (-b * c) * z;
	} 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 = (-y * i) * j
    t_2 = (t * i) * b
    if (b <= (-3.8d+122)) then
        tmp = t_2
    else if (b <= (-4.2d-214)) then
        tmp = t_1
    else if (b <= 9.5d-95) then
        tmp = (x * y) * z
    else if (b <= 0.245d0) then
        tmp = t_1
    else if (b <= 2.9d+187) then
        tmp = (-b * c) * z
    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 = (-y * i) * j;
	double t_2 = (t * i) * b;
	double tmp;
	if (b <= -3.8e+122) {
		tmp = t_2;
	} else if (b <= -4.2e-214) {
		tmp = t_1;
	} else if (b <= 9.5e-95) {
		tmp = (x * y) * z;
	} else if (b <= 0.245) {
		tmp = t_1;
	} else if (b <= 2.9e+187) {
		tmp = (-b * c) * z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-y * i) * j
	t_2 = (t * i) * b
	tmp = 0
	if b <= -3.8e+122:
		tmp = t_2
	elif b <= -4.2e-214:
		tmp = t_1
	elif b <= 9.5e-95:
		tmp = (x * y) * z
	elif b <= 0.245:
		tmp = t_1
	elif b <= 2.9e+187:
		tmp = (-b * c) * z
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-y) * i) * j)
	t_2 = Float64(Float64(t * i) * b)
	tmp = 0.0
	if (b <= -3.8e+122)
		tmp = t_2;
	elseif (b <= -4.2e-214)
		tmp = t_1;
	elseif (b <= 9.5e-95)
		tmp = Float64(Float64(x * y) * z);
	elseif (b <= 0.245)
		tmp = t_1;
	elseif (b <= 2.9e+187)
		tmp = Float64(Float64(Float64(-b) * c) * z);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-y * i) * j;
	t_2 = (t * i) * b;
	tmp = 0.0;
	if (b <= -3.8e+122)
		tmp = t_2;
	elseif (b <= -4.2e-214)
		tmp = t_1;
	elseif (b <= 9.5e-95)
		tmp = (x * y) * z;
	elseif (b <= 0.245)
		tmp = t_1;
	elseif (b <= 2.9e+187)
		tmp = (-b * c) * z;
	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[((-y) * i), $MachinePrecision] * j), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * i), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -3.8e+122], t$95$2, If[LessEqual[b, -4.2e-214], t$95$1, If[LessEqual[b, 9.5e-95], N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 0.245], t$95$1, If[LessEqual[b, 2.9e+187], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.7999999999999998e122 or 2.9000000000000001e187 < b

   1. Initial program 79.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 52.9%

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

   if -3.7999999999999998e122 < b < -4.19999999999999984e-214 or 9.49999999999999998e-95 < b < 0.245

   1. Initial program 77.0%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 53.4

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 40.5%

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

   if -4.19999999999999984e-214 < b < 9.49999999999999998e-95

   1. Initial program 74.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 41.6%

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

   if 0.245 < b < 2.9000000000000001e187

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.6%

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

Alternative 10: 28.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot i\right) \cdot b\\ \mathbf{if}\;b \leq -1.65 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-214}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-35}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* t i) b)))
   (if (<= b -1.65e+79)
     t_1
     (if (<= b -4.2e-214)
       (* (- i) (* j y))
       (if (<= b 8e-35)
         (* (* x y) z)
         (if (<= b 2.9e+187) (* (* (- b) c) 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 = (t * i) * b;
	double tmp;
	if (b <= -1.65e+79) {
		tmp = t_1;
	} else if (b <= -4.2e-214) {
		tmp = -i * (j * y);
	} else if (b <= 8e-35) {
		tmp = (x * y) * z;
	} else if (b <= 2.9e+187) {
		tmp = (-b * c) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * i) * b
    if (b <= (-1.65d+79)) then
        tmp = t_1
    else if (b <= (-4.2d-214)) then
        tmp = -i * (j * y)
    else if (b <= 8d-35) then
        tmp = (x * y) * z
    else if (b <= 2.9d+187) then
        tmp = (-b * c) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (t * i) * b;
	double tmp;
	if (b <= -1.65e+79) {
		tmp = t_1;
	} else if (b <= -4.2e-214) {
		tmp = -i * (j * y);
	} else if (b <= 8e-35) {
		tmp = (x * y) * z;
	} else if (b <= 2.9e+187) {
		tmp = (-b * c) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (t * i) * b
	tmp = 0
	if b <= -1.65e+79:
		tmp = t_1
	elif b <= -4.2e-214:
		tmp = -i * (j * y)
	elif b <= 8e-35:
		tmp = (x * y) * z
	elif b <= 2.9e+187:
		tmp = (-b * c) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(t * i) * b)
	tmp = 0.0
	if (b <= -1.65e+79)
		tmp = t_1;
	elseif (b <= -4.2e-214)
		tmp = Float64(Float64(-i) * Float64(j * y));
	elseif (b <= 8e-35)
		tmp = Float64(Float64(x * y) * z);
	elseif (b <= 2.9e+187)
		tmp = Float64(Float64(Float64(-b) * c) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (t * i) * b;
	tmp = 0.0;
	if (b <= -1.65e+79)
		tmp = t_1;
	elseif (b <= -4.2e-214)
		tmp = -i * (j * y);
	elseif (b <= 8e-35)
		tmp = (x * y) * z;
	elseif (b <= 2.9e+187)
		tmp = (-b * c) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(t * i), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.65e+79], t$95$1, If[LessEqual[b, -4.2e-214], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-35], N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 2.9e+187], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.6500000000000001e79 or 2.9000000000000001e187 < b

   1. Initial program 79.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.3%

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

   if -1.6500000000000001e79 < b < -4.19999999999999984e-214

   1. Initial program 71.1%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in y around inf

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

      1. *-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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.1

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 36.9%

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

   if -4.19999999999999984e-214 < b < 8.00000000000000006e-35

   1. Initial program 76.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.2%

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

   if 8.00000000000000006e-35 < b < 2.9000000000000001e187

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.5%

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

Alternative 11: 50.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+129} \lor \neg \left(b \leq 3.2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.4e+129) (not (<= b 3.2e-11)))
   (* (fma t i (* (- c) z)) 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 <= -1.4e+129) || !(b <= 3.2e-11)) {
		tmp = fma(t, i, (-c * z)) * 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 <= -1.4e+129) || !(b <= 3.2e-11))
		tmp = Float64(fma(t, i, Float64(Float64(-c) * z)) * 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, -1.4e+129], N[Not[LessEqual[b, 3.2e-11]], $MachinePrecision]], N[(N[(t * i + N[((-c) * z), $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 < -1.39999999999999987e129 or 3.19999999999999994e-11 < b

   1. Initial program 77.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 70.9%

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

   if -1.39999999999999987e129 < b < 3.19999999999999994e-11

   1. Initial program 75.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+129} \lor \neg \left(b \leq 3.2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-5} \lor \neg \left(b \leq 5.8 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -5.1e-5) (not (<= b 5.8e-66)))
   (* (fma t i (* (- c) z)) b)
   (* (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 ((b <= -5.1e-5) || !(b <= 5.8e-66)) {
		tmp = fma(t, i, (-c * z)) * b;
	} else {
		tmp = fma(-a, t, (z * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -5.1e-5) || !(b <= 5.8e-66))
		tmp = Float64(fma(t, i, Float64(Float64(-c) * z)) * b);
	else
		tmp = 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[b, -5.1e-5], N[Not[LessEqual[b, 5.8e-66]], $MachinePrecision]], N[(N[(t * i + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.09999999999999996e-5 or 5.80000000000000023e-66 < b

   1. Initial program 77.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 61.9%

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

   if -5.09999999999999996e-5 < b < 5.80000000000000023e-66

   1. Initial program 75.7%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

   8. Applied rewrites 52.8%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-5} \lor \neg \left(b \leq 5.8 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+64}:\\ \;\;\;\;\left(\left(-y\right) \cdot i\right) \cdot j\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -8e+64)
   (* (* (- y) i) j)
   (if (<= y 3.6e+192) (* (fma t i (* (- c) z)) b) (* (- i) (* j 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 (y <= -8e+64) {
		tmp = (-y * i) * j;
	} else if (y <= 3.6e+192) {
		tmp = fma(t, i, (-c * z)) * b;
	} else {
		tmp = -i * (j * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -8e+64)
		tmp = Float64(Float64(Float64(-y) * i) * j);
	elseif (y <= 3.6e+192)
		tmp = Float64(fma(t, i, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(Float64(-i) * Float64(j * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -8e+64], N[(N[((-y) * i), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[y, 3.6e+192], N[(N[(t * i + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000017e64

   1. Initial program 73.2%

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

   3. Taylor expanded in j around -inf

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 60.4

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 55.1%

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

   if -8.00000000000000017e64 < y < 3.6000000000000002e192

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 49.8

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(t, i, \left(-c\right) \cdot z\right) \cdot b \]

   if 3.6000000000000002e192 < y

   1. Initial program 69.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \left(c \cdot j\right) + \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto a \cdot \left(c \cdot j\right) + \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) - b \cdot \left(c \cdot z\right)\right)} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(j \cdot c\right)} - b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(a \cdot j\right) \cdot c} - b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(a \cdot j\right) \cdot c - b \cdot \color{blue}{\left(z \cdot c\right)}\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot j\right) \cdot c - \color{blue}{\left(b \cdot z\right) \cdot c}\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot j - b \cdot z, c, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-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. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 88.3

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.6%

       \[\leadsto \left(-i\right) \cdot \color{blue}{\left(j \cdot y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 29.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+157}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(t \cdot i\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.16e+157)
   (* (* (- c) z) b)
   (if (<= z 4.8e-249)
     (* (* (- i) j) y)
     (if (<= z 2.4e+22) (* (* t i) b) (* (* z x) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.16e+157) {
		tmp = (-c * z) * b;
	} else if (z <= 4.8e-249) {
		tmp = (-i * j) * y;
	} else if (z <= 2.4e+22) {
		tmp = (t * i) * b;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-1.16d+157)) then
        tmp = (-c * z) * b
    else if (z <= 4.8d-249) then
        tmp = (-i * j) * y
    else if (z <= 2.4d+22) then
        tmp = (t * i) * b
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.16e+157) {
		tmp = (-c * z) * b;
	} else if (z <= 4.8e-249) {
		tmp = (-i * j) * y;
	} else if (z <= 2.4e+22) {
		tmp = (t * i) * b;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -1.16e+157:
		tmp = (-c * z) * b
	elif z <= 4.8e-249:
		tmp = (-i * j) * y
	elif z <= 2.4e+22:
		tmp = (t * i) * b
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.16e+157)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (z <= 4.8e-249)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (z <= 2.4e+22)
		tmp = Float64(Float64(t * i) * b);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -1.16e+157)
		tmp = (-c * z) * b;
	elseif (z <= 4.8e-249)
		tmp = (-i * j) * y;
	elseif (z <= 2.4e+22)
		tmp = (t * i) * b;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.16e+157], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[z, 4.8e-249], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.4e+22], N[(N[(t * i), $MachinePrecision] * b), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.16000000000000004e157

   1. Initial program 63.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 70.9

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 59.9%

      \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot b \]

   if -1.16000000000000004e157 < z < 4.80000000000000026e-249

   1. Initial program 82.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \left(c \cdot j\right) + \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto a \cdot \left(c \cdot j\right) + \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) - b \cdot \left(c \cdot z\right)\right)} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(j \cdot c\right)} - b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(a \cdot j\right) \cdot c} - b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(a \cdot j\right) \cdot c - b \cdot \color{blue}{\left(z \cdot c\right)}\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot j\right) \cdot c - \color{blue}{\left(b \cdot z\right) \cdot c}\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot j - b \cdot z, c, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-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. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 44.1%

       \[\leadsto \left(\left(-i\right) \cdot j\right) \cdot y \]

   if 4.80000000000000026e-249 < z < 2.4e22

   1. Initial program 85.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 38.9

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 38.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 28.9%

      \[\leadsto \left(t \cdot i\right) \cdot b \]

   if 2.4e22 < z

   1. Initial program 64.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 67.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.9%

       \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 28.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+124}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+27}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -7.2e+124)
   (* (* z y) x)
   (if (<= x 7e+27) (* (- i) (* j y)) (* (* z x) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -7.2e+124) {
		tmp = (z * y) * x;
	} else if (x <= 7e+27) {
		tmp = -i * (j * y);
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= (-7.2d+124)) then
        tmp = (z * y) * x
    else if (x <= 7d+27) then
        tmp = -i * (j * y)
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -7.2e+124) {
		tmp = (z * y) * x;
	} else if (x <= 7e+27) {
		tmp = -i * (j * y);
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= -7.2e+124:
		tmp = (z * y) * x
	elif x <= 7e+27:
		tmp = -i * (j * y)
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -7.2e+124)
		tmp = Float64(Float64(z * y) * x);
	elseif (x <= 7e+27)
		tmp = Float64(Float64(-i) * Float64(j * y));
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= -7.2e+124)
		tmp = (z * y) * x;
	elseif (x <= 7e+27)
		tmp = -i * (j * y);
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -7.2e+124], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 7e+27], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.19999999999999972e124

   1. Initial program 77.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 60.5

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if -7.19999999999999972e124 < x < 7.0000000000000004e27

   1. Initial program 80.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right) + \left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \left(c \cdot j\right) + \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto a \cdot \left(c \cdot j\right) + \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right)\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) - b \cdot \left(c \cdot z\right)\right)} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(j \cdot c\right)} - b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(a \cdot j\right) \cdot c} - b \cdot \left(c \cdot z\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(a \cdot j\right) \cdot c - b \cdot \color{blue}{\left(z \cdot c\right)}\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot j\right) \cdot c - \color{blue}{\left(b \cdot z\right) \cdot c}\right) + x \cdot \left(y \cdot z - a \cdot t\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} + x \cdot \left(y \cdot z - a \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot j - b \cdot z, c, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-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. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(j \cdot i\right)} + x \cdot z\right) \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right) \cdot i} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, i, x \cdot z\right)} \cdot y \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, i, x \cdot z\right) \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, i, x \cdot z\right) \cdot y \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

      9. lower-*.f64 48.2

         \[\leadsto \mathsf{fma}\left(-j, i, \color{blue}{z \cdot x}\right) \cdot y \]

   8. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, i, z \cdot x\right) \cdot y} \]

   9. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.1%

       \[\leadsto \left(-i\right) \cdot \color{blue}{\left(j \cdot y\right)} \]

   if 7.0000000000000004e27 < x

   1. Initial program 68.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 52.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.5%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.2%

       \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 29.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+52} \lor \neg \left(a \leq 3.1 \cdot 10^{+38}\right):\\ \;\;\;\;\left(c \cdot a\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= a -8.6e+52) (not (<= a 3.1e+38))) (* (* c a) j) (* (* 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 ((a <= -8.6e+52) || !(a <= 3.1e+38)) {
		tmp = (c * a) * j;
	} else {
		tmp = (z * y) * x;
	}
	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 <= (-8.6d+52)) .or. (.not. (a <= 3.1d+38))) then
        tmp = (c * a) * j
    else
        tmp = (z * y) * x
    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 <= -8.6e+52) || !(a <= 3.1e+38)) {
		tmp = (c * a) * j;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (a <= -8.6e+52) or not (a <= 3.1e+38):
		tmp = (c * a) * j
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((a <= -8.6e+52) || !(a <= 3.1e+38))
		tmp = Float64(Float64(c * a) * j);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((a <= -8.6e+52) || ~((a <= 3.1e+38)))
		tmp = (c * a) * j;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[a, -8.6e+52], N[Not[LessEqual[a, 3.1e+38]], $MachinePrecision]], N[(N[(c * a), $MachinePrecision] * j), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5999999999999999e52 or 3.10000000000000018e38 < a

   1. Initial program 65.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(j \cdot \left(-1 \cdot \left(a \cdot c - i \cdot y\right) + -1 \cdot \frac{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - i \cdot t\right)}{j}\right)\right)} \]

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, a, \mathsf{fma}\left(-i, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot t\right), b, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)}{j}\right)\right) \cdot j} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \cdot j \]

      3. mul-1-neg N/A

         \[\leadsto \left(a \cdot c + \color{blue}{-1 \cdot \left(i \cdot y\right)}\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right)} \cdot j \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right) \cdot j} \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot y} + a \cdot c\right) \cdot j \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, a \cdot c\right)} \cdot j \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, a \cdot c\right) \cdot j \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right) \cdot j \]

      10. lower-*.f64 58.4

          \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{a \cdot c}\right) \cdot j \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, a \cdot c\right) \cdot j} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot c\right) \cdot j \]
   10. Step-by-step derivation

   11. Applied rewrites 41.1%

       \[\leadsto \left(c \cdot a\right) \cdot j \]

   if -8.5999999999999999e52 < a < 3.10000000000000018e38

   1. Initial program 82.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 48.4

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.9%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+52} \lor \neg \left(a \leq 3.1 \cdot 10^{+38}\right):\\ \;\;\;\;\left(c \cdot a\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(t \cdot i\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.15e+71)
   (* (* x y) z)
   (if (<= z 2.4e+22) (* (* t i) b) (* (* z x) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.15e+71) {
		tmp = (x * y) * z;
	} else if (z <= 2.4e+22) {
		tmp = (t * i) * b;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-1.15d+71)) then
        tmp = (x * y) * z
    else if (z <= 2.4d+22) then
        tmp = (t * i) * b
    else
        tmp = (z * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -1.15e+71) {
		tmp = (x * y) * z;
	} else if (z <= 2.4e+22) {
		tmp = (t * i) * b;
	} else {
		tmp = (z * x) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -1.15e+71:
		tmp = (x * y) * z
	elif z <= 2.4e+22:
		tmp = (t * i) * b
	else:
		tmp = (z * x) * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.15e+71)
		tmp = Float64(Float64(x * y) * z);
	elseif (z <= 2.4e+22)
		tmp = Float64(Float64(t * i) * b);
	else
		tmp = Float64(Float64(z * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -1.15e+71)
		tmp = (x * y) * z;
	elseif (z <= 2.4e+22)
		tmp = (t * i) * b;
	else
		tmp = (z * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.15e+71], N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 2.4e+22], N[(N[(t * i), $MachinePrecision] * b), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1500000000000001e71

   1. Initial program 70.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 70.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 49.3%

      \[\leadsto \left(x \cdot y\right) \cdot z \]

   if -1.1500000000000001e71 < z < 2.4e22

   1. Initial program 83.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + i \cdot t\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \cdot b \]

      12. remove-double-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \cdot b \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + i \cdot t\right) \cdot b \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + i \cdot t\right) \cdot b \]

      15. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + i \cdot t\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, i \cdot t\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, i \cdot t\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, i \cdot t\right) \cdot b \]

      19. lower-*.f64 40.3

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot t}\right) \cdot b \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 27.2%

      \[\leadsto \left(t \cdot i\right) \cdot b \]

   if 2.4e22 < z

   1. Initial program 64.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

      9. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

      12. lower-*.f64 67.2

          \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.9%

       \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 21.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot y\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* x y) z))

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;
}

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
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;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (x * y) * z

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(x * y) * z)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (x * y) * z;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. *-commutative N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

   9. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   12. lower-*.f64 41.4

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto \left(x \cdot y\right) \cdot z \]
7. Step-by-step derivation

8. Applied rewrites 25.1%

   \[\leadsto \left(x \cdot y\right) \cdot z \]
9. Add Preprocessing

Alternative 19: 21.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot x\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z x) y))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * y;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (z * x) * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * y;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * x) * y

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * x) * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * x) * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. *-commutative N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\left(c \cdot b\right)} + x \cdot y\right) \cdot z \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot c\right) \cdot b} + x \cdot y\right) \cdot z \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot c, b, x \cdot y\right)} \cdot z \]

   9. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(c\right)}, b, x \cdot y\right) \cdot z \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-c}, b, x \cdot y\right) \cdot z \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

   12. lower-*.f64 41.4

       \[\leadsto \mathsf{fma}\left(-c, b, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 24.4%

   \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 25.0%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
11. Add Preprocessing

Developer Target 1: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))