Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 74.0% → 82.2%

Time: 14.1s

Alternatives: 25

Speedup: 0.5×

• 57.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 54.1

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

   5. Applied rewrites 54.1%

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

4. Final simplification 85.6%

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

Alternative 2: 66.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot a\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{if}\;t \leq -3.25 \cdot 10^{+186}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c + \mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot t\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- z) c (* i a)) b (* (fma (- i) j (* z x)) y))))
   (if (<= t -3.25e+186)
     (+ (* (* j t) c) (* (fma (- x) t (* i b)) a))
     (if (<= t -1.35e-6)
       t_1
       (if (<= t -7e-101)
         (fma (fma (- z) b (* j t)) c (* (fma (- t) a (* z y)) x))
         (if (<= t 8e+72) t_1 (* (fma (- x) a (* j c)) t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-z, c, (i * a)), b, (fma(-i, j, (z * x)) * y));
	double tmp;
	if (t <= -3.25e+186) {
		tmp = ((j * t) * c) + (fma(-x, t, (i * b)) * a);
	} else if (t <= -1.35e-6) {
		tmp = t_1;
	} else if (t <= -7e-101) {
		tmp = fma(fma(-z, b, (j * t)), c, (fma(-t, a, (z * y)) * x));
	} else if (t <= 8e+72) {
		tmp = t_1;
	} else {
		tmp = fma(-x, a, (j * c)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-z), c, Float64(i * a)), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y))
	tmp = 0.0
	if (t <= -3.25e+186)
		tmp = Float64(Float64(Float64(j * t) * c) + Float64(fma(Float64(-x), t, Float64(i * b)) * a));
	elseif (t <= -1.35e-6)
		tmp = t_1;
	elseif (t <= -7e-101)
		tmp = fma(fma(Float64(-z), b, Float64(j * t)), c, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	elseif (t <= 8e+72)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-x), a, Float64(j * c)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.25e+186], N[(N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision] + N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.35e-6], t$95$1, If[LessEqual[t, -7e-101], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e+72], t$95$1, N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.2499999999999998e186

   1. Initial program 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 57.3

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 57.3

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 57.2

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

   7. Applied rewrites 57.2%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 78.4

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

   10. Applied rewrites 78.4%

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

   if -3.2499999999999998e186 < t < -1.34999999999999999e-6 or -6.99999999999999989e-101 < t < 7.99999999999999955e72

   1. Initial program 81.8%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   if -1.34999999999999999e-6 < t < -6.99999999999999989e-101

   1. Initial program 71.0%

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

   3. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   if 7.99999999999999955e72 < t

   1. Initial program 69.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.25 \cdot 10^{+186}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c + \mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot a\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot t\right), c, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot a\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y + \left(c \cdot t - i \cdot y\right) \cdot j\\ t_2 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -3 \cdot 10^{+92}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(y - \frac{a \cdot t}{z}\right) \cdot z\right) \cdot x\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (+ (* (* z x) y) (* (- (* c t) (* i y)) j)))
        (t_2 (* (fma (- z) c (* i a)) b)))
   (if (<= b -3e+92)
     t_2
     (if (<= b -7.2e-151)
       t_1
       (if (<= b 1.35e-204)
         (* (* (- y (/ (* a t) z)) z) x)
         (if (<= b 1.18e+83) 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 = ((z * x) * y) + (((c * t) - (i * y)) * j);
	double t_2 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -3e+92) {
		tmp = t_2;
	} else if (b <= -7.2e-151) {
		tmp = t_1;
	} else if (b <= 1.35e-204) {
		tmp = ((y - ((a * t) / z)) * z) * x;
	} else if (b <= 1.18e+83) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(z * x) * y) + Float64(Float64(Float64(c * t) - Float64(i * y)) * j))
	t_2 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -3e+92)
		tmp = t_2;
	elseif (b <= -7.2e-151)
		tmp = t_1;
	elseif (b <= 1.35e-204)
		tmp = Float64(Float64(Float64(y - Float64(Float64(a * t) / z)) * z) * x);
	elseif (b <= 1.18e+83)
		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[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -3e+92], t$95$2, If[LessEqual[b, -7.2e-151], t$95$1, If[LessEqual[b, 1.35e-204], N[(N[(N[(y - N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 1.18e+83], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000013e92 or 1.1799999999999999e83 < b

   1. Initial program 74.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -3.00000000000000013e92 < b < -7.20000000000000064e-151 or 1.34999999999999996e-204 < b < 1.1799999999999999e83

   1. Initial program 76.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -7.20000000000000064e-151 < b < 1.34999999999999996e-204

   1. Initial program 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 79.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 69.5%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 73.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(y - \frac{a \cdot t}{z}\right) \cdot z\right) \cdot x\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{+83}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right)\\ \mathbf{if}\;b \leq -5.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- z) c (* i a))))
   (if (<= b -5.4e+87)
     (fma t_1 b (* (fma (- i) j (* z x)) y))
     (if (<= b 2.4e+84)
       (fma (fma (- t) a (* z y)) x (* (fma (- i) y (* c t)) j))
       (* t_1 b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a));
	double tmp;
	if (b <= -5.4e+87) {
		tmp = fma(t_1, b, (fma(-i, j, (z * x)) * y));
	} else if (b <= 2.4e+84) {
		tmp = fma(fma(-t, a, (z * y)), x, (fma(-i, y, (c * t)) * j));
	} else {
		tmp = t_1 * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-z), c, Float64(i * a))
	tmp = 0.0
	if (b <= -5.4e+87)
		tmp = fma(t_1, b, Float64(fma(Float64(-i), j, Float64(z * x)) * y));
	elseif (b <= 2.4e+84)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-i), y, Float64(c * t)) * j));
	else
		tmp = Float64(t_1 * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.4e+87], N[(t$95$1 * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e+84], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.40000000000000013e87

   1. Initial program 76.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

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

   if -5.40000000000000013e87 < b < 2.4e84

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 76.9

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-fma.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lower-neg.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 80.8

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

   7. Applied rewrites 80.8%

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

   if 2.4e84 < b

   1. Initial program 72.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, c, i \cdot a\right), b, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i a)) b)))
   (if (<= b -8.5e+198)
     t_1
     (if (<= b 1.18e+83)
       (fma (fma (- y) j (* b a)) i (* (fma (- t) a (* z y)) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -8.5e+198) {
		tmp = t_1;
	} else if (b <= 1.18e+83) {
		tmp = fma(fma(-y, j, (b * a)), i, (fma(-t, a, (z * y)) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -8.5e+198)
		tmp = t_1;
	elseif (b <= 1.18e+83)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -8.5e+198], t$95$1, If[LessEqual[b, 1.18e+83], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.5000000000000001e198 or 1.1799999999999999e83 < b

   1. Initial program 74.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -8.5000000000000001e198 < b < 1.1799999999999999e83

   1. Initial program 76.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 69.8%

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

Alternative 6: 58.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c + \mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\left(-z\right) \cdot b\right) \cdot c\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -3.6e-63)
   (+ (* (* j t) c) (* (fma (- x) t (* i b)) a))
   (if (<= a 9.5e-112)
     (fma (fma (- i) y (* c t)) j (* (* (- z) b) c))
     (if (<= a 5.8e+196)
       (fma (* b a) i (* (fma (- t) a (* z y)) x))
       (* (fma (- z) c (* i a)) 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 (a <= -3.6e-63) {
		tmp = ((j * t) * c) + (fma(-x, t, (i * b)) * a);
	} else if (a <= 9.5e-112) {
		tmp = fma(fma(-i, y, (c * t)), j, ((-z * b) * c));
	} else if (a <= 5.8e+196) {
		tmp = fma((b * a), i, (fma(-t, a, (z * y)) * x));
	} else {
		tmp = fma(-z, c, (i * a)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -3.6e-63)
		tmp = Float64(Float64(Float64(j * t) * c) + Float64(fma(Float64(-x), t, Float64(i * b)) * a));
	elseif (a <= 9.5e-112)
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(Float64(Float64(-z) * b) * c));
	elseif (a <= 5.8e+196)
		tmp = fma(Float64(b * a), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	else
		tmp = Float64(fma(Float64(-z), c, Float64(i * a)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -3.6e-63], N[(N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision] + N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e-112], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e+196], N[(N[(b * a), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.60000000000000008e-63

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 71.3

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 70.1

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

   10. Applied rewrites 70.1%

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

   if -3.60000000000000008e-63 < a < 9.50000000000000056e-112

   1. Initial program 78.9%

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

   3. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 71.4

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

   5. Applied rewrites 71.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.4

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 72.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if 9.50000000000000056e-112 < a < 5.8e196

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 84.9

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

   7. Applied rewrites 82.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 78.7%

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

   if 5.8e196 < a

   1. Initial program 52.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c + \mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\left(-z\right) \cdot b\right) \cdot c\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ t_2 := \mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -1 \cdot 10^{+185}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;t \leq 360000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) j (* z x)) y)) (t_2 (* (fma (- x) a (* j c)) t)))
   (if (<= t -1e+185)
     t_2
     (if (<= t -6.5e+89)
       t_1
       (if (<= t 2.1e-301)
         (* (fma y x (* (- b) c)) z)
         (if (<= t 360000000.0) 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(-i, j, (z * x)) * y;
	double t_2 = fma(-x, a, (j * c)) * t;
	double tmp;
	if (t <= -1e+185) {
		tmp = t_2;
	} else if (t <= -6.5e+89) {
		tmp = t_1;
	} else if (t <= 2.1e-301) {
		tmp = fma(y, x, (-b * c)) * z;
	} else if (t <= 360000000.0) {
		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(-i), j, Float64(z * x)) * y)
	t_2 = Float64(fma(Float64(-x), a, Float64(j * c)) * t)
	tmp = 0.0
	if (t <= -1e+185)
		tmp = t_2;
	elseif (t <= -6.5e+89)
		tmp = t_1;
	elseif (t <= 2.1e-301)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	elseif (t <= 360000000.0)
		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[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -1e+185], t$95$2, If[LessEqual[t, -6.5e+89], t$95$1, If[LessEqual[t, 2.1e-301], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 360000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.9999999999999998e184 or 3.6e8 < t

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if -9.9999999999999998e184 < t < -6.4999999999999996e89 or 2.0999999999999999e-301 < t < 3.6e8

   1. Initial program 81.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   if -6.4999999999999996e89 < t < 2.0999999999999999e-301

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 50.1%

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

Alternative 8: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -1.65 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i a)) b)))
   (if (<= b -1.65e-19)
     t_1
     (if (<= b 1.8e+78) (fma (* b a) i (* (fma (- t) a (* z y)) x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -1.65e-19) {
		tmp = t_1;
	} else if (b <= 1.8e+78) {
		tmp = fma((b * a), i, (fma(-t, a, (z * y)) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -1.65e-19)
		tmp = t_1;
	elseif (b <= 1.8e+78)
		tmp = fma(Float64(b * a), i, Float64(fma(Float64(-t), a, Float64(z * y)) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.65e-19], t$95$1, If[LessEqual[b, 1.8e+78], N[(N[(b * a), $MachinePrecision] * i + N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6499999999999999e-19 or 1.8000000000000001e78 < b

   1. Initial program 75.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if -1.6499999999999999e-19 < b < 1.8000000000000001e78

   1. Initial program 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 77.0

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. +-commutative N/A

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 60.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, i, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -1 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(y - \frac{a \cdot t}{z}\right) \cdot z\right) \cdot x\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i a)) b)))
   (if (<= b -1e-59)
     t_1
     (if (<= b 7e-204)
       (* (* (- y (/ (* a t) z)) z) x)
       (if (<= b 9.6e+82) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -1e-59) {
		tmp = t_1;
	} else if (b <= 7e-204) {
		tmp = ((y - ((a * t) / z)) * z) * x;
	} else if (b <= 9.6e+82) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -1e-59)
		tmp = t_1;
	elseif (b <= 7e-204)
		tmp = Float64(Float64(Float64(y - Float64(Float64(a * t) / z)) * z) * x);
	elseif (b <= 9.6e+82)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1e-59], t$95$1, If[LessEqual[b, 7e-204], N[(N[(N[(y - N[(N[(a * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 9.6e+82], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e-59 or 9.59999999999999992e82 < b

   1. Initial program 74.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. *-commutative N/A

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

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

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -1e-59 < b < 7.00000000000000054e-204

   1. Initial program 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 76.4

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 77.4%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 67.3%

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

   if 7.00000000000000054e-204 < b < 9.59999999999999992e82

   1. Initial program 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 78.4

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 62.1

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

   7. Applied rewrites 62.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 67.0%

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

   11. Taylor expanded in j around inf

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

       1. *-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. neg-mul-1 N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-*.f64 53.9

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

   13. Applied rewrites 53.9%

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

Alternative 10: 40.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;b \leq 1.18 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+239}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -1.62e-168)
   (* (fma y x (* (- b) c)) z)
   (if (<= b -5.4e-272)
     (* (* (- x) t) a)
     (if (<= b 1.18e+83)
       (* (fma (- i) j (* z x)) y)
       (if (<= b 2.6e+239) (* (* (- z) b) c) (* (* i b) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.62e-168) {
		tmp = fma(y, x, (-b * c)) * z;
	} else if (b <= -5.4e-272) {
		tmp = (-x * t) * a;
	} else if (b <= 1.18e+83) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (b <= 2.6e+239) {
		tmp = (-z * b) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -1.62e-168)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	elseif (b <= -5.4e-272)
		tmp = Float64(Float64(Float64(-x) * t) * a);
	elseif (b <= 1.18e+83)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (b <= 2.6e+239)
		tmp = Float64(Float64(Float64(-z) * b) * c);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -1.62e-168], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, -5.4e-272], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 1.18e+83], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[b, 2.6e+239], N[(N[((-z) * b), $MachinePrecision] * c), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.6200000000000001e-168

   1. Initial program 74.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 51.9%

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

   if -1.6200000000000001e-168 < b < -5.39999999999999985e-272

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 90.8

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.3%

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

   if -5.39999999999999985e-272 < b < 1.1799999999999999e83

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   if 1.1799999999999999e83 < b < 2.6000000000000002e239

   1. Initial program 80.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 36.0

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

   5. Applied rewrites 36.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.3%

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

   if 2.6000000000000002e239 < b

   1. Initial program 59.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.4%

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

Alternative 11: 30.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot i\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+203}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-164}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* b a) i)))
   (if (<= b -1.1e+203)
     (* (* (- b) c) z)
     (if (<= b -1.2e+81)
       t_1
       (if (<= b -9.5e-164)
         (* (* y x) z)
         (if (<= b 2.1e-21) (* (* (- x) t) 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 = (b * a) * i;
	double tmp;
	if (b <= -1.1e+203) {
		tmp = (-b * c) * z;
	} else if (b <= -1.2e+81) {
		tmp = t_1;
	} else if (b <= -9.5e-164) {
		tmp = (y * x) * z;
	} else if (b <= 2.1e-21) {
		tmp = (-x * t) * a;
	} 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 = (b * a) * i
    if (b <= (-1.1d+203)) then
        tmp = (-b * c) * z
    else if (b <= (-1.2d+81)) then
        tmp = t_1
    else if (b <= (-9.5d-164)) then
        tmp = (y * x) * z
    else if (b <= 2.1d-21) then
        tmp = (-x * t) * a
    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 = (b * a) * i;
	double tmp;
	if (b <= -1.1e+203) {
		tmp = (-b * c) * z;
	} else if (b <= -1.2e+81) {
		tmp = t_1;
	} else if (b <= -9.5e-164) {
		tmp = (y * x) * z;
	} else if (b <= 2.1e-21) {
		tmp = (-x * t) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (b * a) * i
	tmp = 0
	if b <= -1.1e+203:
		tmp = (-b * c) * z
	elif b <= -1.2e+81:
		tmp = t_1
	elif b <= -9.5e-164:
		tmp = (y * x) * z
	elif b <= 2.1e-21:
		tmp = (-x * t) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(b * a) * i)
	tmp = 0.0
	if (b <= -1.1e+203)
		tmp = Float64(Float64(Float64(-b) * c) * z);
	elseif (b <= -1.2e+81)
		tmp = t_1;
	elseif (b <= -9.5e-164)
		tmp = Float64(Float64(y * x) * z);
	elseif (b <= 2.1e-21)
		tmp = Float64(Float64(Float64(-x) * t) * a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (b * a) * i;
	tmp = 0.0;
	if (b <= -1.1e+203)
		tmp = (-b * c) * z;
	elseif (b <= -1.2e+81)
		tmp = t_1;
	elseif (b <= -9.5e-164)
		tmp = (y * x) * z;
	elseif (b <= 2.1e-21)
		tmp = (-x * t) * a;
	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[(b * a), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[b, -1.1e+203], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, -1.2e+81], t$95$1, If[LessEqual[b, -9.5e-164], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 2.1e-21], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.10000000000000002e203

   1. Initial program 77.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.6%

      \[\leadsto \left(\left(-c\right) \cdot b\right) \cdot z \]

   if -1.10000000000000002e203 < b < -1.19999999999999995e81 or 2.10000000000000013e-21 < b

   1. Initial program 73.3%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   if -1.19999999999999995e81 < b < -9.5000000000000001e-164

   1. Initial program 73.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 50.5

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

      \[\leadsto \left(y \cdot x\right) \cdot z \]

   if -9.5000000000000001e-164 < b < 2.10000000000000013e-21

   1. Initial program 79.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 79.5

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 79.5

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 79.5%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 55.9

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.5%

       \[\leadsto \left(\left(-x\right) \cdot t\right) \cdot \color{blue}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+203}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-164}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -1 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i a)) b)))
   (if (<= b -1e-59)
     t_1
     (if (<= b 7e-204)
       (* (fma (- t) a (* z y)) x)
       (if (<= b 9.6e+82) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -1e-59) {
		tmp = t_1;
	} else if (b <= 7e-204) {
		tmp = fma(-t, a, (z * y)) * x;
	} else if (b <= 9.6e+82) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -1e-59)
		tmp = t_1;
	elseif (b <= 7e-204)
		tmp = Float64(fma(Float64(-t), a, Float64(z * y)) * x);
	elseif (b <= 9.6e+82)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1e-59], t$95$1, If[LessEqual[b, 7e-204], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 9.6e+82], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e-59 or 9.59999999999999992e82 < b

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot c + \color{blue}{a \cdot i}\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, a \cdot i\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, a \cdot i\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, a \cdot i\right) \cdot b \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot a}\right) \cdot b \]

      20. lower-*.f64 70.0

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot a}\right) \cdot b \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]

   if -1e-59 < b < 7.00000000000000054e-204

   1. Initial program 76.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 76.4

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 76.4

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{c \cdot \left(j \cdot t\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

      3. lower-*.f64 68.4

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right)} \cdot c \]

   7. Applied rewrites 68.4%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{z} + \left(x \cdot y + \frac{j \cdot \left(c \cdot t - i \cdot y\right)}{z}\right)\right) - \left(-1 \cdot \frac{a \cdot \left(b \cdot i\right)}{z} + b \cdot c\right)\right)} \]

   10. Applied rewrites 77.4%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(t, c, \left(-i\right) \cdot y\right), \frac{j}{z}, y \cdot x\right) + \frac{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a}{z}\right) - c \cdot b\right) \cdot z} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot z - a \cdot t\right) \cdot x} \]

       3. sub-neg N/A

          \[\leadsto \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \cdot x \]

       4. mul-1-neg 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. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot t\right)\right)} + y \cdot z\right) \cdot x \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{t \cdot a}\right)\right) + y \cdot z\right) \cdot x \]

       8. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot a} + y \cdot z\right) \cdot x \]

       9. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot t\right)} \cdot a + y \cdot z\right) \cdot x \]

       10. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot t, a, y \cdot z\right)} \cdot x \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(t\right)}, a, y \cdot z\right) \cdot x \]

       12. lower-neg.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{-t}, a, y \cdot z\right) \cdot x \]

       13. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(-t, a, \color{blue}{z \cdot y}\right) \cdot x \]

       14. lower-*.f64 67.2

           \[\leadsto \mathsf{fma}\left(-t, a, \color{blue}{z \cdot y}\right) \cdot x \]

   13. Applied rewrites 67.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x} \]

   if 7.00000000000000054e-204 < b < 9.59999999999999992e82

   1. Initial program 78.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 78.4

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 78.4

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 78.4%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{c \cdot \left(j \cdot t\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

      3. lower-*.f64 62.1

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right)} \cdot c \]

   7. Applied rewrites 62.1%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{z} + \left(x \cdot y + \frac{j \cdot \left(c \cdot t - i \cdot y\right)}{z}\right)\right) - \left(-1 \cdot \frac{a \cdot \left(b \cdot i\right)}{z} + b \cdot c\right)\right)} \]

   10. Applied rewrites 67.0%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(t, c, \left(-i\right) \cdot y\right), \frac{j}{z}, y \cdot x\right) + \frac{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a}{z}\right) - c \cdot b\right) \cdot z} \]

   11. Taylor expanded in j around inf

       \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

       2. sub-neg N/A

          \[\leadsto \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \cdot j \]

       3. mul-1-neg N/A

          \[\leadsto \left(c \cdot t + \color{blue}{-1 \cdot \left(i \cdot y\right)}\right) \cdot j \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \cdot j \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right) \cdot j} \]

       6. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot y} + c \cdot t\right) \cdot j \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, c \cdot t\right)} \cdot j \]

       8. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

       9. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

       10. lower-*.f64 53.9

           \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   13. Applied rewrites 53.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -1 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i a)) b)))
   (if (<= b -1e-59)
     t_1
     (if (<= b 2e-204)
       (* (fma (- x) a (* j c)) t)
       (if (<= b 9.6e+82) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -1e-59) {
		tmp = t_1;
	} else if (b <= 2e-204) {
		tmp = fma(-x, a, (j * c)) * t;
	} else if (b <= 9.6e+82) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -1e-59)
		tmp = t_1;
	elseif (b <= 2e-204)
		tmp = Float64(fma(Float64(-x), a, Float64(j * c)) * t);
	elseif (b <= 9.6e+82)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1e-59], t$95$1, If[LessEqual[b, 2e-204], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 9.6e+82], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e-59 or 9.59999999999999992e82 < b

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot c + \color{blue}{a \cdot i}\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, a \cdot i\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, a \cdot i\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, a \cdot i\right) \cdot b \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot a}\right) \cdot b \]

      20. lower-*.f64 70.0

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot a}\right) \cdot b \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]

   if -1e-59 < b < 2e-204

   1. Initial program 76.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 55.5

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   if 2e-204 < b < 9.59999999999999992e82

   1. Initial program 78.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 78.4

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 78.4

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 78.4%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{c \cdot \left(j \cdot t\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

      3. lower-*.f64 62.1

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right)} \cdot c \]

   7. Applied rewrites 62.1%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}\right) + \color{blue}{\left(j \cdot t\right) \cdot c} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\left(-1 \cdot \frac{a \cdot \left(t \cdot x\right)}{z} + \left(x \cdot y + \frac{j \cdot \left(c \cdot t - i \cdot y\right)}{z}\right)\right) - \left(-1 \cdot \frac{a \cdot \left(b \cdot i\right)}{z} + b \cdot c\right)\right)} \]

   10. Applied rewrites 67.0%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(t, c, \left(-i\right) \cdot y\right), \frac{j}{z}, y \cdot x\right) + \frac{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a}{z}\right) - c \cdot b\right) \cdot z} \]

   11. Taylor expanded in j around inf

       \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(c \cdot t - i \cdot y\right) \cdot j} \]

       2. sub-neg N/A

          \[\leadsto \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \cdot j \]

       3. mul-1-neg N/A

          \[\leadsto \left(c \cdot t + \color{blue}{-1 \cdot \left(i \cdot y\right)}\right) \cdot j \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \cdot j \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right) \cdot j} \]

       6. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot i\right) \cdot y} + c \cdot t\right) \cdot j \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot i, y, c \cdot t\right)} \cdot j \]

       8. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(i\right)}, y, c \cdot t\right) \cdot j \]

       9. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, y, c \cdot t\right) \cdot j \]

       10. lower-*.f64 53.9

           \[\leadsto \mathsf{fma}\left(-i, y, \color{blue}{c \cdot t}\right) \cdot j \]

   13. Applied rewrites 53.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{if}\;a \leq -4600000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) t (* i b)) a)))
   (if (<= a -4600000000000.0)
     t_1
     (if (<= a 5.6e-111)
       (* (fma (- z) b (* j t)) c)
       (if (<= a 2.35e+14) (* (fma y x (* (- b) c)) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, t, (i * b)) * a;
	double tmp;
	if (a <= -4600000000000.0) {
		tmp = t_1;
	} else if (a <= 5.6e-111) {
		tmp = fma(-z, b, (j * t)) * c;
	} else if (a <= 2.35e+14) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), t, Float64(i * b)) * a)
	tmp = 0.0
	if (a <= -4600000000000.0)
		tmp = t_1;
	elseif (a <= 5.6e-111)
		tmp = Float64(fma(Float64(-z), b, Float64(j * t)) * c);
	elseif (a <= 2.35e+14)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * 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[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -4600000000000.0], t$95$1, If[LessEqual[a, 5.6e-111], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[a, 2.35e+14], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.6e12 or 2.35e14 < a

   1. Initial program 68.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot a \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot t\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot a \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot t} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot a \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot t + \color{blue}{b \cdot i}\right) \cdot a \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, t, b \cdot i\right)} \cdot a \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, t, b \cdot i\right) \cdot a \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, b \cdot i\right) \cdot a \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

      12. lower-*.f64 71.1

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a} \]

   if -4.6e12 < a < 5.5999999999999999e-111

   1. Initial program 80.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot t - b \cdot z\right) \cdot c} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \cdot c \]

      4. mul-1-neg N/A

         \[\leadsto \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \cdot c \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \cdot c \]

      6. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(z \cdot b\right)} + j \cdot t\right) \cdot c \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right) \cdot b} + j \cdot t\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, b, j \cdot t\right)} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, b, j \cdot t\right) \cdot c \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, b, j \cdot t\right) \cdot c \]

      11. lower-*.f64 52.4

          \[\leadsto \mathsf{fma}\left(-z, b, \color{blue}{j \cdot t}\right) \cdot c \]

   5. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c} \]

   if 5.5999999999999999e-111 < a < 2.35e14

   1. Initial program 89.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 54.8

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{if}\;i \leq -9 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -7 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{elif}\;i \leq 6.2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- y) j (* b a)) i)))
   (if (<= i -9e+47)
     t_1
     (if (<= i -7e-202)
       (* (fma (- x) a (* j c)) t)
       (if (<= i 6.2e+90) (* (fma y x (* (- b) c)) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-y, j, (b * a)) * i;
	double tmp;
	if (i <= -9e+47) {
		tmp = t_1;
	} else if (i <= -7e-202) {
		tmp = fma(-x, a, (j * c)) * t;
	} else if (i <= 6.2e+90) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-y), j, Float64(b * a)) * i)
	tmp = 0.0
	if (i <= -9e+47)
		tmp = t_1;
	elseif (i <= -7e-202)
		tmp = Float64(fma(Float64(-x), a, Float64(j * c)) * t);
	elseif (i <= 6.2e+90)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * 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[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -9e+47], t$95$1, If[LessEqual[i, -7e-202], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[i, 6.2e+90], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if i < -8.99999999999999958e47 or 6.19999999999999977e90 < i

   1. Initial program 66.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 70.6

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   if -8.99999999999999958e47 < i < -6.9999999999999998e-202

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 58.1

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   if -6.9999999999999998e-202 < i < 6.19999999999999977e90

   1. Initial program 84.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 48.9

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.9%

      \[\leadsto \mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 30.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+25}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-267}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-118}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+18}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -6e+25)
   (* (* b a) i)
   (if (<= a 1.75e-267)
     (* (* j c) t)
     (if (<= a 1.45e-118)
       (* (* (- z) b) c)
       (if (<= a 4.5e+18) (* (* z y) x) (* (* i b) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -6e+25) {
		tmp = (b * a) * i;
	} else if (a <= 1.75e-267) {
		tmp = (j * c) * t;
	} else if (a <= 1.45e-118) {
		tmp = (-z * b) * c;
	} else if (a <= 4.5e+18) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-6d+25)) then
        tmp = (b * a) * i
    else if (a <= 1.75d-267) then
        tmp = (j * c) * t
    else if (a <= 1.45d-118) then
        tmp = (-z * b) * c
    else if (a <= 4.5d+18) then
        tmp = (z * y) * x
    else
        tmp = (i * b) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -6e+25) {
		tmp = (b * a) * i;
	} else if (a <= 1.75e-267) {
		tmp = (j * c) * t;
	} else if (a <= 1.45e-118) {
		tmp = (-z * b) * c;
	} else if (a <= 4.5e+18) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -6e+25:
		tmp = (b * a) * i
	elif a <= 1.75e-267:
		tmp = (j * c) * t
	elif a <= 1.45e-118:
		tmp = (-z * b) * c
	elif a <= 4.5e+18:
		tmp = (z * y) * x
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -6e+25)
		tmp = Float64(Float64(b * a) * i);
	elseif (a <= 1.75e-267)
		tmp = Float64(Float64(j * c) * t);
	elseif (a <= 1.45e-118)
		tmp = Float64(Float64(Float64(-z) * b) * c);
	elseif (a <= 4.5e+18)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -6e+25)
		tmp = (b * a) * i;
	elseif (a <= 1.75e-267)
		tmp = (j * c) * t;
	elseif (a <= 1.45e-118)
		tmp = (-z * b) * c;
	elseif (a <= 4.5e+18)
		tmp = (z * y) * x;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -6e+25], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[a, 1.75e-267], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[a, 1.45e-118], N[(N[((-z) * b), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[a, 4.5e+18], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -6.00000000000000011e25

   1. Initial program 64.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 51.3

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 40.8%

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   if -6.00000000000000011e25 < a < 1.75e-267

   1. Initial program 82.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 82.0

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 82.0

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 82.0%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 42.1

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 35.5%

       \[\leadsto \left(j \cdot c\right) \cdot t \]

   if 1.75e-267 < a < 1.4499999999999999e-118

   1. Initial program 76.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 44.9

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \left(\left(-z\right) \cdot b\right) \cdot \color{blue}{c} \]

   if 1.4499999999999999e-118 < a < 4.5e18

   1. Initial program 89.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 51.4

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]

   if 4.5e18 < a

   1. Initial program 70.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 47.0

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.9%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{+25}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-267}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-118}:\\ \;\;\;\;\left(\left(-z\right) \cdot b\right) \cdot c\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+18}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(j \cdot c\right) \cdot t\\ t_2 := \left(b \cdot a\right) \cdot i\\ \mathbf{if}\;j \leq -1.5 \cdot 10^{+270}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;j \leq -5.3 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-238}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* j c) t)) (t_2 (* (* b a) i)))
   (if (<= j -1.5e+270)
     t_2
     (if (<= j -5.3e+110)
       t_1
       (if (<= j 7.8e-238) t_2 (if (<= j 6.5e-42) (* (* 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 = (j * c) * t;
	double t_2 = (b * a) * i;
	double tmp;
	if (j <= -1.5e+270) {
		tmp = t_2;
	} else if (j <= -5.3e+110) {
		tmp = t_1;
	} else if (j <= 7.8e-238) {
		tmp = t_2;
	} else if (j <= 6.5e-42) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (j * c) * t
    t_2 = (b * a) * i
    if (j <= (-1.5d+270)) then
        tmp = t_2
    else if (j <= (-5.3d+110)) then
        tmp = t_1
    else if (j <= 7.8d-238) then
        tmp = t_2
    else if (j <= 6.5d-42) then
        tmp = (y * x) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (j * c) * t;
	double t_2 = (b * a) * i;
	double tmp;
	if (j <= -1.5e+270) {
		tmp = t_2;
	} else if (j <= -5.3e+110) {
		tmp = t_1;
	} else if (j <= 7.8e-238) {
		tmp = t_2;
	} else if (j <= 6.5e-42) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (j * c) * t
	t_2 = (b * a) * i
	tmp = 0
	if j <= -1.5e+270:
		tmp = t_2
	elif j <= -5.3e+110:
		tmp = t_1
	elif j <= 7.8e-238:
		tmp = t_2
	elif j <= 6.5e-42:
		tmp = (y * x) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(j * c) * t)
	t_2 = Float64(Float64(b * a) * i)
	tmp = 0.0
	if (j <= -1.5e+270)
		tmp = t_2;
	elseif (j <= -5.3e+110)
		tmp = t_1;
	elseif (j <= 7.8e-238)
		tmp = t_2;
	elseif (j <= 6.5e-42)
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (j * c) * t;
	t_2 = (b * a) * i;
	tmp = 0.0;
	if (j <= -1.5e+270)
		tmp = t_2;
	elseif (j <= -5.3e+110)
		tmp = t_1;
	elseif (j <= 7.8e-238)
		tmp = t_2;
	elseif (j <= 6.5e-42)
		tmp = (y * x) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[j, -1.5e+270], t$95$2, If[LessEqual[j, -5.3e+110], t$95$1, If[LessEqual[j, 7.8e-238], t$95$2, If[LessEqual[j, 6.5e-42], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.50000000000000007e270 or -5.2999999999999998e110 < j < 7.7999999999999997e-238

   1. Initial program 70.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 48.0

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 33.6%

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   if -1.50000000000000007e270 < j < -5.2999999999999998e110 or 6.4999999999999998e-42 < j

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 79.7

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 79.7

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 79.7%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 55.7

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 43.8%

       \[\leadsto \left(j \cdot c\right) \cdot t \]

   if 7.7999999999999997e-238 < j < 6.4999999999999998e-42

   1. Initial program 83.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 67.7

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

      \[\leadsto \left(y \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.5 \cdot 10^{+270}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq -5.3 \cdot 10^{+110}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-238}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{+221}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+162}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -6.1e+221)
   (* (* j t) c)
   (if (<= t 4.4e+66)
     (* (fma y x (* (- b) c)) z)
     (if (<= t 1.1e+162) (* (* j c) t) (* (* (- x) t) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -6.1e+221) {
		tmp = (j * t) * c;
	} else if (t <= 4.4e+66) {
		tmp = fma(y, x, (-b * c)) * z;
	} else if (t <= 1.1e+162) {
		tmp = (j * c) * t;
	} else {
		tmp = (-x * t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -6.1e+221)
		tmp = Float64(Float64(j * t) * c);
	elseif (t <= 4.4e+66)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	elseif (t <= 1.1e+162)
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = Float64(Float64(Float64(-x) * t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -6.1e+221], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[t, 4.4e+66], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t, 1.1e+162], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.0999999999999998e221

   1. Initial program 56.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 56.7

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 56.7

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 56.7%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.1%

       \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]

   if -6.0999999999999998e221 < t < 4.3999999999999997e66

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 44.7

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.7%

      \[\leadsto \mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z \]

   if 4.3999999999999997e66 < t < 1.1000000000000001e162

   1. Initial program 75.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 75.1

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 75.1

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 75.1%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 75.9

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot t \]
   9. Step-by-step derivation

   10. Applied rewrites 57.7%

       \[\leadsto \left(j \cdot c\right) \cdot t \]

   if 1.1000000000000001e162 < t

   1. Initial program 67.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 67.7

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 67.7

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 67.7%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 80.8

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.7%

       \[\leadsto \left(\left(-x\right) \cdot t\right) \cdot \color{blue}{a} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{if}\;b \leq -1 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- z) c (* i a)) b)))
   (if (<= b -1e-59) t_1 (if (<= b 8.6e+43) (* (fma (- x) a (* j c)) t) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-z, c, (i * a)) * b;
	double tmp;
	if (b <= -1e-59) {
		tmp = t_1;
	} else if (b <= 8.6e+43) {
		tmp = fma(-x, a, (j * c)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-z), c, Float64(i * a)) * b)
	tmp = 0.0
	if (b <= -1e-59)
		tmp = t_1;
	elseif (b <= 8.6e+43)
		tmp = Float64(fma(Float64(-x), a, Float64(j * c)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1e-59], t$95$1, If[LessEqual[b, 8.6e+43], N[(N[((-x) * a + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1e-59 or 8.6e43 < b

   1. Initial program 76.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \cdot b \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right)} \cdot b \]

      4. remove-double-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)}\right) \cdot b \]

      5. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - a \cdot i\right)}\right)\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right)} \cdot b \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right)} \cdot b \]

      10. sub-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(a \cdot i\right)\right)\right)}\right)\right) \cdot b \]

      11. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right)} \cdot b \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot c}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot c} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      14. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot z\right)} \cdot c + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(-1 \cdot z\right) \cdot c + \color{blue}{a \cdot i}\right) \cdot b \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot z, c, a \cdot i\right)} \cdot b \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(z\right)}, c, a \cdot i\right) \cdot b \]

      18. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-z}, c, a \cdot i\right) \cdot b \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot a}\right) \cdot b \]

      20. lower-*.f64 67.4

          \[\leadsto \mathsf{fma}\left(-z, c, \color{blue}{i \cdot a}\right) \cdot b \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]

   if -1e-59 < b < 8.6e43

   1. Initial program 76.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 52.3

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) t (* i b)) a)))
   (if (<= a -4.1e-89)
     t_1
     (if (<= a 2.35e+14) (* (fma y x (* (- b) c)) z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-x, t, (i * b)) * a;
	double tmp;
	if (a <= -4.1e-89) {
		tmp = t_1;
	} else if (a <= 2.35e+14) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), t, Float64(i * b)) * a)
	tmp = 0.0
	if (a <= -4.1e-89)
		tmp = t_1;
	elseif (a <= 2.35e+14)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * 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[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -4.1e-89], t$95$1, If[LessEqual[a, 2.35e+14], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.0999999999999998e-89 or 2.35e14 < a

   1. Initial program 72.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \cdot a \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot t\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot a \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot t} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \cdot a \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot x\right) \cdot t + \color{blue}{b \cdot i}\right) \cdot a \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, t, b \cdot i\right)} \cdot a \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, t, b \cdot i\right) \cdot a \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, b \cdot i\right) \cdot a \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

      12. lower-*.f64 66.4

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a} \]

   if -4.0999999999999998e-89 < a < 2.35e14

   1. Initial program 80.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 45.2

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.1%

      \[\leadsto \mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(j \cdot t\right) \cdot c\\ \mathbf{if}\;j \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-238}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* j t) c)))
   (if (<= j -3.4e+24)
     t_1
     (if (<= j 7.8e-238)
       (* (* b a) i)
       (if (<= j 6.5e-42) (* (* 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 = (j * t) * c;
	double tmp;
	if (j <= -3.4e+24) {
		tmp = t_1;
	} else if (j <= 7.8e-238) {
		tmp = (b * a) * i;
	} else if (j <= 6.5e-42) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (j * t) * c
    if (j <= (-3.4d+24)) then
        tmp = t_1
    else if (j <= 7.8d-238) then
        tmp = (b * a) * i
    else if (j <= 6.5d-42) then
        tmp = (y * x) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (j * t) * c;
	double tmp;
	if (j <= -3.4e+24) {
		tmp = t_1;
	} else if (j <= 7.8e-238) {
		tmp = (b * a) * i;
	} else if (j <= 6.5e-42) {
		tmp = (y * x) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (j * t) * c
	tmp = 0
	if j <= -3.4e+24:
		tmp = t_1
	elif j <= 7.8e-238:
		tmp = (b * a) * i
	elif j <= 6.5e-42:
		tmp = (y * x) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(j * t) * c)
	tmp = 0.0
	if (j <= -3.4e+24)
		tmp = t_1;
	elseif (j <= 7.8e-238)
		tmp = Float64(Float64(b * a) * i);
	elseif (j <= 6.5e-42)
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (j * t) * c;
	tmp = 0.0;
	if (j <= -3.4e+24)
		tmp = t_1;
	elseif (j <= 7.8e-238)
		tmp = (b * a) * i;
	elseif (j <= 6.5e-42)
		tmp = (y * x) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[j, -3.4e+24], t$95$1, If[LessEqual[j, 7.8e-238], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[j, 6.5e-42], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.4000000000000001e24 or 6.4999999999999998e-42 < j

   1. Initial program 74.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\frac{1}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. un-div-inv N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{c \cdot z + i \cdot a}{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      7. clear-num N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{\frac{\left(c \cdot z\right) \cdot \left(c \cdot z\right) - \left(i \cdot a\right) \cdot \left(i \cdot a\right)}{c \cdot z + i \cdot a}}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      8. flip-- N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      10. lower-/.f64 74.6

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\color{blue}{\frac{1}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z - i \cdot a}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      12. sub-neg N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{c \cdot z + \left(\mathsf{neg}\left(i \cdot a\right)\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      13. +-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(i \cdot a\right)\right) + c \cdot z}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{i \cdot a}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{a \cdot i}\right)\right) + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot i} + c \cdot z}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      18. lower-neg.f64 74.6

          \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b}{\frac{1}{\mathsf{fma}\left(\color{blue}{-a}, i, c \cdot z\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   4. Applied rewrites 74.6%

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\frac{b}{\frac{1}{\mathsf{fma}\left(-a, i, c \cdot z\right)}}}\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(x \cdot a\right)} + c \cdot j\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right) \cdot a} + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot x, a, c \cdot j\right)} \cdot t \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(x\right)}, a, c \cdot j\right) \cdot t \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, a, c \cdot j\right) \cdot t \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

      9. lower-*.f64 48.0

         \[\leadsto \mathsf{fma}\left(-x, a, \color{blue}{j \cdot c}\right) \cdot t \]

   7. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, a, j \cdot c\right) \cdot t} \]

   8. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.8%

       \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]

   if -3.4000000000000001e24 < j < 7.7999999999999997e-238

   1. Initial program 74.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 46.2

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 33.6%

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   if 7.7999999999999997e-238 < j < 6.4999999999999998e-42

   1. Initial program 83.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 67.7

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

      \[\leadsto \left(y \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{-238}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+83}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -1.3e+83)
   (* (* i b) a)
   (if (<= b 4.2e+44) (* (* z y) x) (* (* b a) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.3e+83) {
		tmp = (i * b) * a;
	} else if (b <= 4.2e+44) {
		tmp = (z * y) * x;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-1.3d+83)) then
        tmp = (i * b) * a
    else if (b <= 4.2d+44) then
        tmp = (z * y) * x
    else
        tmp = (b * a) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -1.3e+83) {
		tmp = (i * b) * a;
	} else if (b <= 4.2e+44) {
		tmp = (z * y) * x;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -1.3e+83:
		tmp = (i * b) * a
	elif b <= 4.2e+44:
		tmp = (z * y) * x
	else:
		tmp = (b * a) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -1.3e+83)
		tmp = Float64(Float64(i * b) * a);
	elseif (b <= 4.2e+44)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(b * a) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -1.3e+83)
		tmp = (i * b) * a;
	elseif (b <= 4.2e+44)
		tmp = (z * y) * x;
	else
		tmp = (b * a) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -1.3e+83], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 4.2e+44], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3000000000000001e83

   1. Initial program 75.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 47.9

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   if -1.3000000000000001e83 < b < 4.19999999999999974e44

   1. Initial program 76.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 32.8

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]

   if 4.19999999999999974e44 < b

   1. Initial program 76.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 52.4

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 44.7%

      \[\leadsto \left(a \cdot b\right) \cdot i \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+83}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 29.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot b\right) \cdot a\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+45}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* i b) a)))
   (if (<= b -1.3e+83) t_1 (if (<= b 1.2e+45) (* (* z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * b) * a;
	double tmp;
	if (b <= -1.3e+83) {
		tmp = t_1;
	} else if (b <= 1.2e+45) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (i * b) * a
    if (b <= (-1.3d+83)) then
        tmp = t_1
    else if (b <= 1.2d+45) then
        tmp = (z * y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (i * b) * a;
	double tmp;
	if (b <= -1.3e+83) {
		tmp = t_1;
	} else if (b <= 1.2e+45) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (i * b) * a
	tmp = 0
	if b <= -1.3e+83:
		tmp = t_1
	elif b <= 1.2e+45:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(i * b) * a)
	tmp = 0.0
	if (b <= -1.3e+83)
		tmp = t_1;
	elseif (b <= 1.2e+45)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (i * b) * a;
	tmp = 0.0;
	if (b <= -1.3e+83)
		tmp = t_1;
	elseif (b <= 1.2e+45)
		tmp = (z * y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, -1.3e+83], t$95$1, If[LessEqual[b, 1.2e+45], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3000000000000001e83 or 1.19999999999999995e45 < b

   1. Initial program 75.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \cdot i \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \cdot i \]

      7. remove-double-neg N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 50.0

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.0%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   if -1.3000000000000001e83 < b < 1.19999999999999995e45

   1. Initial program 76.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

      11. lower-*.f64 32.8

          \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   5. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+83}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+45}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 22.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot y\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z y) x))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (z * y) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * y) * x;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * y) * x

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 76.1%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   11. lower-*.f64 37.0

       \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 37.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 20.8%

   \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]

9. Final simplification 20.8%

   \[\leadsto \left(z \cdot y\right) \cdot x \]
10. Add Preprocessing

Alternative 25: 22.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot x\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* z x) y))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * y;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (z * x) * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (z * x) * y;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (z * x) * y

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(z * x) * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (z * x) * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 76.1%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y - b \cdot c\right) \cdot z} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \cdot z \]

   4. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \cdot z \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \cdot z \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \cdot z \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot b, c, x \cdot y\right)} \cdot z \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(b\right)}, c, x \cdot y\right) \cdot z \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-b}, c, x \cdot y\right) \cdot z \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

   11. lower-*.f64 37.0

       \[\leadsto \mathsf{fma}\left(-b, c, \color{blue}{y \cdot x}\right) \cdot z \]

5. Applied rewrites 37.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
7. Step-by-step derivation

8. Applied rewrites 20.8%

   \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 19.2%

    \[\leadsto \left(x \cdot z\right) \cdot y \]

11. Final simplification 19.2%

    \[\leadsto \left(z \cdot x\right) \cdot y \]
12. Add Preprocessing

Developer Target 1: 69.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024294 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))