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

Percentage Accurate: 72.9% → 82.1%

Time: 15.0s

Alternatives: 22

Speedup: 0.5×

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

Specification

Math

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

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: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.1% accurate, 0.4× speedup

Math

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

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double 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(-j, y, ((fma(fma(-z, b, (j * a)), c, (fma(-a, t, (z * y)) * x)) / i) + (b * t))) * i;
	}
	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(-j), y, Float64(Float64(fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-a), t, Float64(z * y)) * x)) / i) + Float64(b * t))) * i);
	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[((-j) * y + N[(N[(N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(b * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.2%

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

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

   5. Applied rewrites 49.2%

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

Alternative 2: 81.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 49.2%

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

Alternative 3: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(-b, \mathsf{fma}\left(-i, t, c \cdot z\right), \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* i b)) t)))
   (if (<= t -3.8e+171)
     t_1
     (if (<= t 6e-304)
       (fma (fma (- z) b (* j a)) c (* (fma (- a) t (* z y)) x))
       (if (<= t 2.05e+85)
         (fma (fma (- i) y (* c a)) j (* (fma (- b) c (* y x)) z))
         (if (<= t 1.9e+204)
           (fma (- b) (fma (- i) t (* c z)) (* (fma (- x) t (* j c)) a))
           t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, x, (i * b)) * t;
	double tmp;
	if (t <= -3.8e+171) {
		tmp = t_1;
	} else if (t <= 6e-304) {
		tmp = fma(fma(-z, b, (j * a)), c, (fma(-a, t, (z * y)) * x));
	} else if (t <= 2.05e+85) {
		tmp = fma(fma(-i, y, (c * a)), j, (fma(-b, c, (y * x)) * z));
	} else if (t <= 1.9e+204) {
		tmp = fma(-b, fma(-i, t, (c * z)), (fma(-x, t, (j * c)) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), x, Float64(i * b)) * t)
	tmp = 0.0
	if (t <= -3.8e+171)
		tmp = t_1;
	elseif (t <= 6e-304)
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	elseif (t <= 2.05e+85)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	elseif (t <= 1.9e+204)
		tmp = fma(Float64(-b), fma(Float64(-i), t, Float64(c * z)), Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -3.8e+171], t$95$1, If[LessEqual[t, 6e-304], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e+85], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e+204], N[((-b) * N[((-i) * t + N[(c * z), $MachinePrecision]), $MachinePrecision] + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.8000000000000002e171 or 1.8999999999999999e204 < t

   1. Initial program 59.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -3.8000000000000002e171 < t < 6.0000000000000002e-304

   1. Initial program 74.3%

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

   3. Taylor expanded in i around 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. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 76.5%

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

   if 6.0000000000000002e-304 < t < 2.04999999999999989e85

   1. Initial program 81.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 80.5%

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

   if 2.04999999999999989e85 < t < 1.8999999999999999e204

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. *-commutative N/A

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

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

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

      19. lower-fma.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

Alternative 4: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot a\right), c, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* i b)) t)))
   (if (<= t -3.8e+171)
     t_1
     (if (<= t 6e-304)
       (fma (fma (- z) b (* j a)) c (* (fma (- a) t (* z y)) x))
       (if (<= t 1.5e+87)
         (fma (fma (- i) y (* c a)) j (* (fma (- b) c (* y x)) z))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, x, (i * b)) * t;
	double tmp;
	if (t <= -3.8e+171) {
		tmp = t_1;
	} else if (t <= 6e-304) {
		tmp = fma(fma(-z, b, (j * a)), c, (fma(-a, t, (z * y)) * x));
	} else if (t <= 1.5e+87) {
		tmp = fma(fma(-i, y, (c * a)), j, (fma(-b, c, (y * x)) * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), x, Float64(i * b)) * t)
	tmp = 0.0
	if (t <= -3.8e+171)
		tmp = t_1;
	elseif (t <= 6e-304)
		tmp = fma(fma(Float64(-z), b, Float64(j * a)), c, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	elseif (t <= 1.5e+87)
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -3.8e+171], t$95$1, If[LessEqual[t, 6e-304], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e+87], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.8000000000000002e171 or 1.4999999999999999e87 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if -3.8000000000000002e171 < t < 6.0000000000000002e-304

   1. Initial program 74.3%

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

   3. Taylor expanded in i around 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. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 76.5%

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

   if 6.0000000000000002e-304 < t < 1.4999999999999999e87

   1. Initial program 81.8%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 80.5%

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

Alternative 5: 51.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ t_2 := \mathsf{fma}\left(-t, a, y \cdot z\right) \cdot x\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-75}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t + \left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 425:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma a c (* (- i) y)) j)) (t_2 (* (fma (- t) a (* y z)) x)))
   (if (<= x -4.8e+64)
     t_2
     (if (<= x -5.8e-75)
       (+ (* (* i b) t) (* (* j c) a))
       (if (<= x -3.6e-191)
         (* (fma j a (* (- z) b)) c)
         (if (<= x 1.8e-267)
           t_1
           (if (<= x 425.0)
             (* (fma (- z) c (* i t)) b)
             (if (<= x 5.4e+89) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(a, c, (-i * y)) * j;
	double t_2 = fma(-t, a, (y * z)) * x;
	double tmp;
	if (x <= -4.8e+64) {
		tmp = t_2;
	} else if (x <= -5.8e-75) {
		tmp = ((i * b) * t) + ((j * c) * a);
	} else if (x <= -3.6e-191) {
		tmp = fma(j, a, (-z * b)) * c;
	} else if (x <= 1.8e-267) {
		tmp = t_1;
	} else if (x <= 425.0) {
		tmp = fma(-z, c, (i * t)) * b;
	} else if (x <= 5.4e+89) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(a, c, Float64(Float64(-i) * y)) * j)
	t_2 = Float64(fma(Float64(-t), a, Float64(y * z)) * x)
	tmp = 0.0
	if (x <= -4.8e+64)
		tmp = t_2;
	elseif (x <= -5.8e-75)
		tmp = Float64(Float64(Float64(i * b) * t) + Float64(Float64(j * c) * a));
	elseif (x <= -3.6e-191)
		tmp = Float64(fma(j, a, Float64(Float64(-z) * b)) * c);
	elseif (x <= 1.8e-267)
		tmp = t_1;
	elseif (x <= 425.0)
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	elseif (x <= 5.4e+89)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t) * a + N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.8e+64], t$95$2, If[LessEqual[x, -5.8e-75], N[(N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision] + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.6e-191], N[(N[(j * a + N[((-z) * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 1.8e-267], t$95$1, If[LessEqual[x, 425.0], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 5.4e+89], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -4.79999999999999999e64 or 5.4e89 < x

   1. Initial program 71.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 70.0%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

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

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

       3. mul-1-neg N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. *-commutative N/A

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

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

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

       10. mul-1-neg N/A

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

       11. lower-fma.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-*.f64 70.1

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

   11. Applied rewrites 70.1%

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

   if -4.79999999999999999e64 < x < -5.8000000000000003e-75

   1. Initial program 68.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in i around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if -5.8000000000000003e-75 < x < -3.60000000000000019e-191

   1. Initial program 65.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. mul-1-neg N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. *-commutative N/A

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

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

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

       11. mul-1-neg N/A

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

       12. lower-*.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if -3.60000000000000019e-191 < x < 1.8000000000000001e-267 or 425 < x < 5.4e89

   1. Initial program 83.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 79.6%

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

   9. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if 1.8000000000000001e-267 < x < 425

   1. Initial program 71.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 6: 68.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{+86} \lor \neg \left(t \leq 1.5 \cdot 10^{+87}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -7.4e+86) (not (<= t 1.5e+87)))
   (* (fma (- a) x (* i b)) t)
   (fma (fma (- i) y (* c a)) j (* (fma (- b) c (* y x)) z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -7.4e+86) || !(t <= 1.5e+87)) {
		tmp = fma(-a, x, (i * b)) * t;
	} else {
		tmp = fma(fma(-i, y, (c * a)), j, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -7.4e+86) || !(t <= 1.5e+87))
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * a)), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -7.4e+86], N[Not[LessEqual[t, 1.5e+87]], $MachinePrecision]], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.39999999999999983e86 or 1.4999999999999999e87 < t

   1. Initial program 66.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -7.39999999999999983e86 < t < 1.4999999999999999e87

   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 t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 73.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{+86} \lor \neg \left(t \leq 1.5 \cdot 10^{+87}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ t_2 := \mathsf{fma}\left(-t, a, y \cdot z\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 425:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot t\right) \cdot b\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma a c (* (- i) y)) j)) (t_2 (* (fma (- t) a (* y z)) x)))
   (if (<= x -2.3e+74)
     t_2
     (if (<= x -3.6e-191)
       (* (fma j a (* (- z) b)) c)
       (if (<= x 1.8e-267)
         t_1
         (if (<= x 425.0)
           (* (fma (- z) c (* i t)) b)
           (if (<= x 5.4e+89) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(a, c, (-i * y)) * j;
	double t_2 = fma(-t, a, (y * z)) * x;
	double tmp;
	if (x <= -2.3e+74) {
		tmp = t_2;
	} else if (x <= -3.6e-191) {
		tmp = fma(j, a, (-z * b)) * c;
	} else if (x <= 1.8e-267) {
		tmp = t_1;
	} else if (x <= 425.0) {
		tmp = fma(-z, c, (i * t)) * b;
	} else if (x <= 5.4e+89) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(a, c, Float64(Float64(-i) * y)) * j)
	t_2 = Float64(fma(Float64(-t), a, Float64(y * z)) * x)
	tmp = 0.0
	if (x <= -2.3e+74)
		tmp = t_2;
	elseif (x <= -3.6e-191)
		tmp = Float64(fma(j, a, Float64(Float64(-z) * b)) * c);
	elseif (x <= 1.8e-267)
		tmp = t_1;
	elseif (x <= 425.0)
		tmp = Float64(fma(Float64(-z), c, Float64(i * t)) * b);
	elseif (x <= 5.4e+89)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t) * a + N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e+74], t$95$2, If[LessEqual[x, -3.6e-191], N[(N[(j * a + N[((-z) * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 1.8e-267], t$95$1, If[LessEqual[x, 425.0], N[(N[((-z) * c + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 5.4e+89], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.2999999999999999e74 or 5.4e89 < x

   1. Initial program 73.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

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

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

       3. mul-1-neg N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. *-commutative N/A

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

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

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

       10. mul-1-neg N/A

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

       11. lower-fma.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-*.f64 70.5

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

   11. Applied rewrites 70.5%

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

   if -2.2999999999999999e74 < x < -3.60000000000000019e-191

   1. Initial program 65.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. mul-1-neg N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. *-commutative N/A

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

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

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

       11. mul-1-neg N/A

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

       12. lower-*.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 51.7

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

   11. Applied rewrites 51.7%

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

   if -3.60000000000000019e-191 < x < 1.8000000000000001e-267 or 425 < x < 5.4e89

   1. Initial program 83.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 79.6%

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

   9. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if 1.8000000000000001e-267 < x < 425

   1. Initial program 71.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   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. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 8: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.12 \cdot 10^{+92} \lor \neg \left(z \leq 1.65 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(c \cdot a, j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -2.12e+92) (not (<= z 1.65e-5)))
   (fma (* c a) j (* (fma (- b) c (* y x)) z))
   (+ (* (* i t) b) (* 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) {
	double tmp;
	if ((z <= -2.12e+92) || !(z <= 1.65e-5)) {
		tmp = fma((c * a), j, (fma(-b, c, (y * x)) * z));
	} else {
		tmp = ((i * t) * b) + (j * ((c * a) - (y * i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -2.12e+92) || !(z <= 1.65e-5))
		tmp = fma(Float64(c * a), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	else
		tmp = Float64(Float64(Float64(i * t) * b) + Float64(j * Float64(Float64(c * a) - Float64(y * i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -2.12e+92], N[Not[LessEqual[z, 1.65e-5]], $MachinePrecision]], N[(N[(c * a), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.11999999999999999e92 or 1.6500000000000001e-5 < z

   1. Initial program 65.5%

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

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.5%

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

   if -2.11999999999999999e92 < z < 1.6500000000000001e-5

   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 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 63.1

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

   5. Applied rewrites 63.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.12 \cdot 10^{+92} \lor \neg \left(z \leq 1.65 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(c \cdot a, j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, y \cdot z\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot a\right) \cdot c\\ \mathbf{elif}\;x \leq 19000000000000:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* y z)) x)))
   (if (<= x -2.3e+74)
     t_1
     (if (<= x 4.2e-79)
       (* (fma (- z) b (* j a)) c)
       (if (<= x 19000000000000.0)
         (* (fma (- a) x (* i b)) t)
         (if (<= x 5.4e+89) (* (fma a c (* (- i) y)) j) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (y * z)) * x;
	double tmp;
	if (x <= -2.3e+74) {
		tmp = t_1;
	} else if (x <= 4.2e-79) {
		tmp = fma(-z, b, (j * a)) * c;
	} else if (x <= 19000000000000.0) {
		tmp = fma(-a, x, (i * b)) * t;
	} else if (x <= 5.4e+89) {
		tmp = fma(a, c, (-i * y)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(y * z)) * x)
	tmp = 0.0
	if (x <= -2.3e+74)
		tmp = t_1;
	elseif (x <= 4.2e-79)
		tmp = Float64(fma(Float64(-z), b, Float64(j * a)) * c);
	elseif (x <= 19000000000000.0)
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	elseif (x <= 5.4e+89)
		tmp = Float64(fma(a, c, Float64(Float64(-i) * y)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e+74], t$95$1, If[LessEqual[x, 4.2e-79], N[(N[((-z) * b + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 19000000000000.0], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, 5.4e+89], N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.2999999999999999e74 or 5.4e89 < x

   1. Initial program 73.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

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

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

       3. mul-1-neg N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. *-commutative N/A

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

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

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

       10. mul-1-neg N/A

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

       11. lower-fma.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-*.f64 70.5

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

   11. Applied rewrites 70.5%

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

   if -2.2999999999999999e74 < x < 4.1999999999999999e-79

   1. Initial program 71.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   if 4.1999999999999999e-79 < x < 1.9e13

   1. Initial program 65.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if 1.9e13 < x < 5.4e89

   1. Initial program 88.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 78.5

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

   11. Applied rewrites 78.5%

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

Alternative 10: 51.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, y \cdot z\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;x \leq 19000000000000:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* y z)) x)))
   (if (<= x -2.3e+74)
     t_1
     (if (<= x 4.2e-79)
       (* (fma j a (* (- z) b)) c)
       (if (<= x 19000000000000.0)
         (* (fma (- a) x (* i b)) t)
         (if (<= x 5.4e+89) (* (fma a c (* (- i) y)) j) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (y * z)) * x;
	double tmp;
	if (x <= -2.3e+74) {
		tmp = t_1;
	} else if (x <= 4.2e-79) {
		tmp = fma(j, a, (-z * b)) * c;
	} else if (x <= 19000000000000.0) {
		tmp = fma(-a, x, (i * b)) * t;
	} else if (x <= 5.4e+89) {
		tmp = fma(a, c, (-i * y)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(y * z)) * x)
	tmp = 0.0
	if (x <= -2.3e+74)
		tmp = t_1;
	elseif (x <= 4.2e-79)
		tmp = Float64(fma(j, a, Float64(Float64(-z) * b)) * c);
	elseif (x <= 19000000000000.0)
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	elseif (x <= 5.4e+89)
		tmp = Float64(fma(a, c, Float64(Float64(-i) * y)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e+74], t$95$1, If[LessEqual[x, 4.2e-79], N[(N[(j * a + N[((-z) * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 19000000000000.0], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, 5.4e+89], N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.2999999999999999e74 or 5.4e89 < x

   1. Initial program 73.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

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

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

       3. mul-1-neg N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. *-commutative N/A

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

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

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

       10. mul-1-neg N/A

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

       11. lower-fma.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-*.f64 70.5

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

   11. Applied rewrites 70.5%

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

   if -2.2999999999999999e74 < x < 4.1999999999999999e-79

   1. Initial program 71.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. mul-1-neg N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. *-commutative N/A

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

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

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

       11. mul-1-neg N/A

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

       12. lower-*.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 54.1

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

   11. Applied rewrites 54.1%

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

   if 4.1999999999999999e-79 < x < 1.9e13

   1. Initial program 65.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if 1.9e13 < x < 5.4e89

   1. Initial program 88.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 78.5

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

   11. Applied rewrites 78.5%

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

Alternative 11: 52.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ t_2 := \mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -4.4 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;j \leq -8 \cdot 10^{-154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.95 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;j \leq 3.5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* i b)) t)) (t_2 (* (fma a c (* (- i) y)) j)))
   (if (<= j -4.4e+130)
     t_2
     (if (<= j -8e-154)
       t_1
       (if (<= j 2.95e-218)
         (* (fma (- b) c (* y x)) z)
         (if (<= j 3.5e+16) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, x, (i * b)) * t;
	double t_2 = fma(a, c, (-i * y)) * j;
	double tmp;
	if (j <= -4.4e+130) {
		tmp = t_2;
	} else if (j <= -8e-154) {
		tmp = t_1;
	} else if (j <= 2.95e-218) {
		tmp = fma(-b, c, (y * x)) * z;
	} else if (j <= 3.5e+16) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), x, Float64(i * b)) * t)
	t_2 = Float64(fma(a, c, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -4.4e+130)
		tmp = t_2;
	elseif (j <= -8e-154)
		tmp = t_1;
	elseif (j <= 2.95e-218)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	elseif (j <= 3.5e+16)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -4.4e+130], t$95$2, If[LessEqual[j, -8e-154], t$95$1, If[LessEqual[j, 2.95e-218], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[j, 3.5e+16], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if j < -4.39999999999999987e130 or 3.5e16 < j

   1. Initial program 74.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 70.2

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

   11. Applied rewrites 70.2%

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

   if -4.39999999999999987e130 < j < -7.9999999999999998e-154 or 2.95000000000000003e-218 < j < 3.5e16

   1. Initial program 73.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if -7.9999999999999998e-154 < j < 2.95000000000000003e-218

   1. Initial program 68.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

Alternative 12: 42.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -66000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -1.5 \cdot 10^{-153}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq -1.72 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;j \leq 4.4 \cdot 10^{-8}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma a c (* (- i) y)) j)))
   (if (<= j -66000000000.0)
     t_1
     (if (<= j -1.5e-153)
       (* (* i b) t)
       (if (<= j -1.72e-196)
         (* (* (- c) z) b)
         (if (<= j 4.4e-8) (* (- a) (* t x)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(a, c, (-i * y)) * j;
	double tmp;
	if (j <= -66000000000.0) {
		tmp = t_1;
	} else if (j <= -1.5e-153) {
		tmp = (i * b) * t;
	} else if (j <= -1.72e-196) {
		tmp = (-c * z) * b;
	} else if (j <= 4.4e-8) {
		tmp = -a * (t * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(a, c, Float64(Float64(-i) * y)) * j)
	tmp = 0.0
	if (j <= -66000000000.0)
		tmp = t_1;
	elseif (j <= -1.5e-153)
		tmp = Float64(Float64(i * b) * t);
	elseif (j <= -1.72e-196)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (j <= 4.4e-8)
		tmp = Float64(Float64(-a) * Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -66000000000.0], t$95$1, If[LessEqual[j, -1.5e-153], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, -1.72e-196], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[j, 4.4e-8], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -6.6e10 or 4.3999999999999997e-8 < j

   1. Initial program 72.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in j around inf

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

   8. Applied rewrites 83.6%

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

   9. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 62.2

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

   11. Applied rewrites 62.2%

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

   if -6.6e10 < j < -1.5e-153

   1. Initial program 68.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.6%

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

   if -1.5e-153 < j < -1.72e-196

   1. Initial program 64.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 75.6

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 75.8%

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

   if -1.72e-196 < j < 4.3999999999999997e-8

   1. Initial program 76.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.7%

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

Alternative 13: 61.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+85} \lor \neg \left(t \leq 2.25 \cdot 10^{+83}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a, j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -7.2e+85) (not (<= t 2.25e+83)))
   (* (fma (- a) x (* i b)) t)
   (fma (* c a) j (* (fma (- b) c (* y x)) z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -7.2e+85) || !(t <= 2.25e+83)) {
		tmp = fma(-a, x, (i * b)) * t;
	} else {
		tmp = fma((c * a), j, (fma(-b, c, (y * x)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -7.2e+85) || !(t <= 2.25e+83))
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	else
		tmp = fma(Float64(c * a), j, Float64(fma(Float64(-b), c, Float64(y * x)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -7.2e+85], N[Not[LessEqual[t, 2.25e+83]], $MachinePrecision]], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(c * a), $MachinePrecision] * j + N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.1999999999999996e85 or 2.25e83 < t

   1. Initial program 66.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -7.1999999999999996e85 < t < 2.25e83

   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 t around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

      6. lower-fma.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+85} \lor \neg \left(t \leq 2.25 \cdot 10^{+83}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a, j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot c\right) \cdot j\\ \mathbf{if}\;j \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -1.5 \cdot 10^{-153}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq -1.72 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;j \leq 9.6 \cdot 10^{+16}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* a c) j)))
   (if (<= j -1.75e+16)
     t_1
     (if (<= j -1.5e-153)
       (* (* i b) t)
       (if (<= j -1.72e-196)
         (* (* (- c) z) b)
         (if (<= j 9.6e+16) (* (- a) (* t x)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -1.5e-153) {
		tmp = (i * b) * t;
	} else if (j <= -1.72e-196) {
		tmp = (-c * z) * b;
	} else if (j <= 9.6e+16) {
		tmp = -a * (t * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * c) * j
    if (j <= (-1.75d+16)) then
        tmp = t_1
    else if (j <= (-1.5d-153)) then
        tmp = (i * b) * t
    else if (j <= (-1.72d-196)) then
        tmp = (-c * z) * b
    else if (j <= 9.6d+16) then
        tmp = -a * (t * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -1.5e-153) {
		tmp = (i * b) * t;
	} else if (j <= -1.72e-196) {
		tmp = (-c * z) * b;
	} else if (j <= 9.6e+16) {
		tmp = -a * (t * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (a * c) * j
	tmp = 0
	if j <= -1.75e+16:
		tmp = t_1
	elif j <= -1.5e-153:
		tmp = (i * b) * t
	elif j <= -1.72e-196:
		tmp = (-c * z) * b
	elif j <= 9.6e+16:
		tmp = -a * (t * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * c) * j)
	tmp = 0.0
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -1.5e-153)
		tmp = Float64(Float64(i * b) * t);
	elseif (j <= -1.72e-196)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (j <= 9.6e+16)
		tmp = Float64(Float64(-a) * Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (a * c) * j;
	tmp = 0.0;
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -1.5e-153)
		tmp = (i * b) * t;
	elseif (j <= -1.72e-196)
		tmp = (-c * z) * b;
	elseif (j <= 9.6e+16)
		tmp = -a * (t * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.75e+16], t$95$1, If[LessEqual[j, -1.5e-153], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, -1.72e-196], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[j, 9.6e+16], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.75e16 or 9.6e16 < j

   1. Initial program 71.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 54.5

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.6%

       \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.7%

       \[\leadsto \left(a \cdot c\right) \cdot j \]

   if -1.75e16 < j < -1.5e-153

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 59.9

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 45.6%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -1.5e-153 < j < -1.72e-196

   1. Initial program 64.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 75.6

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.8%

       \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot \color{blue}{b} \]

   if -1.72e-196 < j < 9.6e16

   1. Initial program 77.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 47.6

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.1%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(t \cdot x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* i b)) t)))
   (if (<= t -7.5e-18)
     t_1
     (if (<= t -4.2e-300)
       (* (fma j a (* (- z) b)) c)
       (if (<= t 2.05e+85) (* (fma a c (* (- i) y)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, x, (i * b)) * t;
	double tmp;
	if (t <= -7.5e-18) {
		tmp = t_1;
	} else if (t <= -4.2e-300) {
		tmp = fma(j, a, (-z * b)) * c;
	} else if (t <= 2.05e+85) {
		tmp = fma(a, c, (-i * y)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), x, Float64(i * b)) * t)
	tmp = 0.0
	if (t <= -7.5e-18)
		tmp = t_1;
	elseif (t <= -4.2e-300)
		tmp = Float64(fma(j, a, Float64(Float64(-z) * b)) * c);
	elseif (t <= 2.05e+85)
		tmp = Float64(fma(a, c, Float64(Float64(-i) * y)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -7.5e-18], t$95$1, If[LessEqual[t, -4.2e-300], N[(N[(j * a + N[((-z) * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t, 2.05e+85], N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.50000000000000015e-18 or 2.04999999999999989e85 < t

   1. Initial program 68.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 62.3

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   if -7.50000000000000015e-18 < t < -4.20000000000000007e-300

   1. Initial program 68.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(\left(a \cdot c + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{j}\right) - \left(i \cdot y + \frac{b \cdot \left(c \cdot z - i \cdot t\right)}{j}\right)\right)} \]

   8. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c, a, x \cdot \frac{\mathsf{fma}\left(-t, a, z \cdot y\right)}{j}\right) - \mathsf{fma}\left(i, y, b \cdot \frac{\mathsf{fma}\left(c, z, \left(-i\right) \cdot t\right)}{j}\right)\right) \cdot j} \]

   9. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b\right)\right) \cdot z\right)} \cdot c \]

       4. mul-1-neg N/A

          \[\leadsto \left(a \cdot j + \color{blue}{\left(-1 \cdot b\right)} \cdot z\right) \cdot c \]

       5. associate-*r* N/A

          \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

       6. *-commutative N/A

          \[\leadsto \left(\color{blue}{j \cdot a} + -1 \cdot \left(b \cdot z\right)\right) \cdot c \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(j, a, -1 \cdot \left(b \cdot z\right)\right)} \cdot c \]

       8. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \cdot c \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(j, a, \mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) \cdot c \]

       10. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot b}\right) \cdot c \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(-1 \cdot z\right)} \cdot b\right) \cdot c \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(-1 \cdot z\right) \cdot b}\right) \cdot c \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot b\right) \cdot c \]

       14. lower-neg.f64 63.0

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(-z\right)} \cdot b\right) \cdot c \]

   11. Applied rewrites 63.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c} \]

   if -4.20000000000000007e-300 < t < 2.04999999999999989e85

   1. Initial program 82.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(\left(a \cdot c + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{j}\right) - \left(i \cdot y + \frac{b \cdot \left(c \cdot z - i \cdot t\right)}{j}\right)\right)} \]

   8. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c, a, x \cdot \frac{\mathsf{fma}\left(-t, a, z \cdot y\right)}{j}\right) - \mathsf{fma}\left(i, y, b \cdot \frac{\mathsf{fma}\left(c, z, \left(-i\right) \cdot t\right)}{j}\right)\right) \cdot j} \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, c, \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y}\right) \cdot j \]

       6. lower-neg.f64 52.3

          \[\leadsto \mathsf{fma}\left(a, c, \color{blue}{\left(-i\right)} \cdot y\right) \cdot j \]

   11. Applied rewrites 52.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- a) (* t x))))
   (if (<= x -2.25e+135)
     t_1
     (if (<= x 1.15e-89)
       (* (fma j a (* (- z) b)) c)
       (if (<= x 1.2e+121) (* (fma a c (* (- i) y)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -a * (t * x);
	double tmp;
	if (x <= -2.25e+135) {
		tmp = t_1;
	} else if (x <= 1.15e-89) {
		tmp = fma(j, a, (-z * b)) * c;
	} else if (x <= 1.2e+121) {
		tmp = fma(a, c, (-i * y)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(-a) * Float64(t * x))
	tmp = 0.0
	if (x <= -2.25e+135)
		tmp = t_1;
	elseif (x <= 1.15e-89)
		tmp = Float64(fma(j, a, Float64(Float64(-z) * b)) * c);
	elseif (x <= 1.2e+121)
		tmp = Float64(fma(a, c, Float64(Float64(-i) * y)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.25e+135], t$95$1, If[LessEqual[x, 1.15e-89], N[(N[(j * a + N[((-z) * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 1.2e+121], N[(N[(a * c + N[((-i) * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.25000000000000004e135 or 1.2e121 < x

   1. Initial program 73.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 54.5

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(t \cdot x\right)} \]

   if -2.25000000000000004e135 < x < 1.15e-89

   1. Initial program 71.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(\left(a \cdot c + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{j}\right) - \left(i \cdot y + \frac{b \cdot \left(c \cdot z - i \cdot t\right)}{j}\right)\right)} \]

   8. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c, a, x \cdot \frac{\mathsf{fma}\left(-t, a, z \cdot y\right)}{j}\right) - \mathsf{fma}\left(i, y, b \cdot \frac{\mathsf{fma}\left(c, z, \left(-i\right) \cdot t\right)}{j}\right)\right) \cdot j} \]

   9. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot j - b \cdot z\right) \cdot c} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b\right)\right) \cdot z\right)} \cdot c \]

       4. mul-1-neg N/A

          \[\leadsto \left(a \cdot j + \color{blue}{\left(-1 \cdot b\right)} \cdot z\right) \cdot c \]

       5. associate-*r* N/A

          \[\leadsto \left(a \cdot j + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

       6. *-commutative N/A

          \[\leadsto \left(\color{blue}{j \cdot a} + -1 \cdot \left(b \cdot z\right)\right) \cdot c \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(j, a, -1 \cdot \left(b \cdot z\right)\right)} \cdot c \]

       8. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\mathsf{neg}\left(b \cdot z\right)}\right) \cdot c \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(j, a, \mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) \cdot c \]

       10. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot b}\right) \cdot c \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(-1 \cdot z\right)} \cdot b\right) \cdot c \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(-1 \cdot z\right) \cdot b}\right) \cdot c \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot b\right) \cdot c \]

       14. lower-neg.f64 52.5

           \[\leadsto \mathsf{fma}\left(j, a, \color{blue}{\left(-z\right)} \cdot b\right) \cdot c \]

   11. Applied rewrites 52.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(j, a, \left(-z\right) \cdot b\right) \cdot c} \]

   if 1.15e-89 < x < 1.2e121

   1. Initial program 75.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(\left(a \cdot c + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{j}\right) - \left(i \cdot y + \frac{b \cdot \left(c \cdot z - i \cdot t\right)}{j}\right)\right)} \]

   8. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c, a, x \cdot \frac{\mathsf{fma}\left(-t, a, z \cdot y\right)}{j}\right) - \mathsf{fma}\left(i, y, b \cdot \frac{\mathsf{fma}\left(c, z, \left(-i\right) \cdot t\right)}{j}\right)\right) \cdot j} \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, \left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \cdot j \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, c, \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y}\right) \cdot j \]

       6. lower-neg.f64 55.2

          \[\leadsto \mathsf{fma}\left(a, c, \color{blue}{\left(-i\right)} \cdot y\right) \cdot j \]

   11. Applied rewrites 55.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, \left(-i\right) \cdot y\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot c\right) \cdot j\\ \mathbf{if}\;j \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -3.2 \cdot 10^{-121}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 9.6 \cdot 10^{+16}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* a c) j)))
   (if (<= j -1.75e+16)
     t_1
     (if (<= j -3.2e-121)
       (* (* i b) t)
       (if (<= j 9.6e+16) (* (- a) (* t x)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -3.2e-121) {
		tmp = (i * b) * t;
	} else if (j <= 9.6e+16) {
		tmp = -a * (t * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * c) * j
    if (j <= (-1.75d+16)) then
        tmp = t_1
    else if (j <= (-3.2d-121)) then
        tmp = (i * b) * t
    else if (j <= 9.6d+16) then
        tmp = -a * (t * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -3.2e-121) {
		tmp = (i * b) * t;
	} else if (j <= 9.6e+16) {
		tmp = -a * (t * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (a * c) * j
	tmp = 0
	if j <= -1.75e+16:
		tmp = t_1
	elif j <= -3.2e-121:
		tmp = (i * b) * t
	elif j <= 9.6e+16:
		tmp = -a * (t * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * c) * j)
	tmp = 0.0
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -3.2e-121)
		tmp = Float64(Float64(i * b) * t);
	elseif (j <= 9.6e+16)
		tmp = Float64(Float64(-a) * Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (a * c) * j;
	tmp = 0.0;
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -3.2e-121)
		tmp = (i * b) * t;
	elseif (j <= 9.6e+16)
		tmp = -a * (t * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.75e+16], t$95$1, If[LessEqual[j, -3.2e-121], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 9.6e+16], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.75e16 or 9.6e16 < j

   1. Initial program 71.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 54.5

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.6%

       \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.7%

       \[\leadsto \left(a \cdot c\right) \cdot j \]

   if -1.75e16 < j < -3.20000000000000019e-121

   1. Initial program 61.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 48.0%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -3.20000000000000019e-121 < j < 9.6e16

   1. Initial program 77.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 47.2

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.4%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(t \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 29.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot c\right) \cdot j\\ \mathbf{if}\;j \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -7.2 \cdot 10^{-121}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 9.6 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* a c) j)))
   (if (<= j -1.75e+16)
     t_1
     (if (<= j -7.2e-121)
       (* (* i b) t)
       (if (<= j 9.6e+16) (* (* (- t) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -7.2e-121) {
		tmp = (i * b) * t;
	} else if (j <= 9.6e+16) {
		tmp = (-t * a) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * c) * j
    if (j <= (-1.75d+16)) then
        tmp = t_1
    else if (j <= (-7.2d-121)) then
        tmp = (i * b) * t
    else if (j <= 9.6d+16) then
        tmp = (-t * a) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -7.2e-121) {
		tmp = (i * b) * t;
	} else if (j <= 9.6e+16) {
		tmp = (-t * a) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (a * c) * j
	tmp = 0
	if j <= -1.75e+16:
		tmp = t_1
	elif j <= -7.2e-121:
		tmp = (i * b) * t
	elif j <= 9.6e+16:
		tmp = (-t * a) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * c) * j)
	tmp = 0.0
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -7.2e-121)
		tmp = Float64(Float64(i * b) * t);
	elseif (j <= 9.6e+16)
		tmp = Float64(Float64(Float64(-t) * a) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (a * c) * j;
	tmp = 0.0;
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -7.2e-121)
		tmp = (i * b) * t;
	elseif (j <= 9.6e+16)
		tmp = (-t * a) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.75e+16], t$95$1, If[LessEqual[j, -7.2e-121], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 9.6e+16], N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.75e16 or 9.6e16 < j

   1. Initial program 71.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 54.5

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.6%

       \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.7%

       \[\leadsto \left(a \cdot c\right) \cdot j \]

   if -1.75e16 < j < -7.19999999999999967e-121

   1. Initial program 61.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 48.0%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -7.19999999999999967e-121 < j < 9.6e16

   1. Initial program 77.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 47.2

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot t\right) + \frac{b \cdot \left(i \cdot t\right)}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \mathsf{fma}\left(-t, a, b \cdot \frac{i \cdot t}{x}\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot t\right)\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 33.6%

       \[\leadsto \left(\left(-t\right) \cdot a\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 29.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot c\right) \cdot j\\ \mathbf{if}\;j \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -7.6 \cdot 10^{-122}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;j \leq 1.56 \cdot 10^{-16}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* a c) j)))
   (if (<= j -1.75e+16)
     t_1
     (if (<= j -7.6e-122)
       (* (* i b) t)
       (if (<= j 1.56e-16) (* (* z y) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -7.6e-122) {
		tmp = (i * b) * t;
	} else if (j <= 1.56e-16) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * c) * j
    if (j <= (-1.75d+16)) then
        tmp = t_1
    else if (j <= (-7.6d-122)) then
        tmp = (i * b) * t
    else if (j <= 1.56d-16) then
        tmp = (z * y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -7.6e-122) {
		tmp = (i * b) * t;
	} else if (j <= 1.56e-16) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (a * c) * j
	tmp = 0
	if j <= -1.75e+16:
		tmp = t_1
	elif j <= -7.6e-122:
		tmp = (i * b) * t
	elif j <= 1.56e-16:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * c) * j)
	tmp = 0.0
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -7.6e-122)
		tmp = Float64(Float64(i * b) * t);
	elseif (j <= 1.56e-16)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (a * c) * j;
	tmp = 0.0;
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -7.6e-122)
		tmp = (i * b) * t;
	elseif (j <= 1.56e-16)
		tmp = (z * y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.75e+16], t$95$1, If[LessEqual[j, -7.6e-122], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 1.56e-16], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.75e16 or 1.55999999999999996e-16 < j

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 52.5

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.2%

       \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 43.1%

       \[\leadsto \left(a \cdot c\right) \cdot j \]

   if -1.75e16 < j < -7.6000000000000002e-122

   1. Initial program 61.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(b \cdot i\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 48.0%

      \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -7.6000000000000002e-122 < j < 1.55999999999999996e-16

   1. Initial program 76.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right) \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, z \cdot \left(x \cdot y - b \cdot c\right)\right)} \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, a \cdot c\right)}, j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z}\right) \]

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)} \]

   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 27.1%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot c\right) \cdot j\\ \mathbf{if}\;j \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -7.6 \cdot 10^{-122}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;j \leq 1.56 \cdot 10^{-16}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* a c) j)))
   (if (<= j -1.75e+16)
     t_1
     (if (<= j -7.6e-122)
       (* (* i t) b)
       (if (<= j 1.56e-16) (* (* z y) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -7.6e-122) {
		tmp = (i * t) * b;
	} else if (j <= 1.56e-16) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * c) * j
    if (j <= (-1.75d+16)) then
        tmp = t_1
    else if (j <= (-7.6d-122)) then
        tmp = (i * t) * b
    else if (j <= 1.56d-16) then
        tmp = (z * y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (a * c) * j;
	double tmp;
	if (j <= -1.75e+16) {
		tmp = t_1;
	} else if (j <= -7.6e-122) {
		tmp = (i * t) * b;
	} else if (j <= 1.56e-16) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (a * c) * j
	tmp = 0
	if j <= -1.75e+16:
		tmp = t_1
	elif j <= -7.6e-122:
		tmp = (i * t) * b
	elif j <= 1.56e-16:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(a * c) * j)
	tmp = 0.0
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -7.6e-122)
		tmp = Float64(Float64(i * t) * b);
	elseif (j <= 1.56e-16)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (a * c) * j;
	tmp = 0.0;
	if (j <= -1.75e+16)
		tmp = t_1;
	elseif (j <= -7.6e-122)
		tmp = (i * t) * b;
	elseif (j <= 1.56e-16)
		tmp = (z * y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.75e+16], t$95$1, If[LessEqual[j, -7.6e-122], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[j, 1.56e-16], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.75e16 or 1.55999999999999996e-16 < j

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 52.5

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.2%

       \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 43.1%

       \[\leadsto \left(a \cdot c\right) \cdot j \]

   if -1.75e16 < j < -7.6000000000000002e-122

   1. Initial program 61.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.6%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]

   if -7.6000000000000002e-122 < j < 1.55999999999999996e-16

   1. Initial program 76.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\left(j \cdot \left(a \cdot c - i \cdot y\right) + x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot c - i \cdot y\right) \cdot j} + \left(x \cdot \left(y \cdot z\right) - b \cdot \left(c \cdot z\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(\color{blue}{\left(x \cdot y\right) \cdot z} - b \cdot \left(c \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \left(\left(x \cdot y\right) \cdot z - \color{blue}{\left(b \cdot c\right) \cdot z}\right) \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j + \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot c - i \cdot y, j, z \cdot \left(x \cdot y - b \cdot c\right)\right)} \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot c + \left(\mathsf{neg}\left(i\right)\right) \cdot y}, j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot y + a \cdot c}, j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, a \cdot c\right)}, j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-i}, y, a \cdot c\right), j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, \color{blue}{c \cdot a}\right), j, z \cdot \left(x \cdot y - b \cdot c\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z}\right) \]

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\right)} \]

   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 27.1%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.75 \cdot 10^{+16} \lor \neg \left(j \leq 165000000000\right):\\ \;\;\;\;\left(a \cdot c\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= j -1.75e+16) (not (<= j 165000000000.0)))
   (* (* a c) j)
   (* (* 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 ((j <= -1.75e+16) || !(j <= 165000000000.0)) {
		tmp = (a * c) * j;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((j <= (-1.75d+16)) .or. (.not. (j <= 165000000000.0d0))) then
        tmp = (a * c) * j
    else
        tmp = (i * t) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -1.75e+16) || !(j <= 165000000000.0)) {
		tmp = (a * c) * j;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (j <= -1.75e+16) or not (j <= 165000000000.0):
		tmp = (a * c) * j
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((j <= -1.75e+16) || !(j <= 165000000000.0))
		tmp = Float64(Float64(a * c) * j);
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((j <= -1.75e+16) || ~((j <= 165000000000.0)))
		tmp = (a * c) * j;
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[j, -1.75e+16], N[Not[LessEqual[j, 165000000000.0]], $MachinePrecision]], N[(N[(a * c), $MachinePrecision] * j), $MachinePrecision], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -1.75e16 or 1.65e11 < j

   1. Initial program 72.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(i \cdot t + \left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right)\right) - c \cdot z\right) \cdot b} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(i \cdot t + \left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - c \cdot z\right) + i \cdot t\right)} \cdot b \]

      4. associate--r- N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right)} \cdot b \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{j \cdot \left(a \cdot c - i \cdot y\right)}{b} + \frac{x \cdot \left(y \cdot z - a \cdot t\right)}{b}\right) - \left(c \cdot z - i \cdot t\right)\right) \cdot b} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-c, z, \mathsf{fma}\left(i, t, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot a\right), j, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)}{b}\right)\right) \cdot b} \]

   6. Taylor expanded in c around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(c \cdot \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot c}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(\mathsf{neg}\left(c\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \color{blue}{\left(-1 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) - -1 \cdot \left(b \cdot z\right)\right) \cdot \left(-1 \cdot c\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(-1 \cdot b\right) \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot j\right) - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot j\right) + b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot j\right)\right)} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{j \cdot a}\right)\right) + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(j\right)\right) \cdot a} + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      12. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot j\right)} \cdot a + b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot j, a, b \cdot z\right)} \cdot \left(-1 \cdot c\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(j\right)}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-j}, a, b \cdot z\right) \cdot \left(-1 \cdot c\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, a, \color{blue}{b \cdot z}\right) \cdot \left(-1 \cdot c\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \]

      18. lower-neg.f64 54.1

          \[\leadsto \mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \color{blue}{\left(-c\right)} \]

   8. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, a, b \cdot z\right) \cdot \left(-c\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.3%

       \[\leadsto \left(j \cdot c\right) \cdot \color{blue}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.4%

       \[\leadsto \left(a \cdot c\right) \cdot j \]

   if -1.75e16 < j < 1.65e11

   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 t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

      4. metadata-eval N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. *-lft-identity N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

      11. lower-*.f64 48.5

          \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.75 \cdot 10^{+16} \lor \neg \left(j \leq 165000000000\right):\\ \;\;\;\;\left(a \cdot c\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot t\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i t) b))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * t) * b;
}

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 = (i * t) * b
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 (i * t) * b;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * t) * b

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * t) * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * t) * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right)} \cdot t \]

   4. metadata-eval N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot t \]

   5. *-lft-identity N/A

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + \color{blue}{b \cdot i}\right) \cdot t \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + b \cdot i\right) \cdot t \]

   7. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + b \cdot i\right) \cdot t \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right)} \cdot t \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, b \cdot i\right) \cdot t \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

   11. lower-*.f64 39.4

       \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{i \cdot b}\right) \cdot t \]

5. Applied rewrites 39.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]

6. Taylor expanded in x around 0

   \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.9%

   \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
9. Add Preprocessing

Developer Target 1: 58.9% 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 2024320 
(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)))))