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

Percentage Accurate: 74.3% → 82.4%

Time: 15.5s

Alternatives: 16

Speedup: 0.5×

• 58.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in i around -inf

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

   8. Applied rewrites 57.3%

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

4. Final simplification 84.1%

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

Alternative 2: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ t_2 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* i b)) t)) (t_2 (* (fma (- c) b (* y x)) z)))
   (if (<= z -6.8e+34)
     t_2
     (if (<= z -2.2e-193)
       t_1
       (if (<= z 1.3e-297)
         (* (fma (- i) y (* c a)) j)
         (if (<= z 3.8e-195)
           t_1
           (if (<= z 5.5e+15)
             (fma (* b t) i (* (fma (- b) z (* j a)) c))
             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(-x, a, (i * b)) * t;
	double t_2 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -6.8e+34) {
		tmp = t_2;
	} else if (z <= -2.2e-193) {
		tmp = t_1;
	} else if (z <= 1.3e-297) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (z <= 3.8e-195) {
		tmp = t_1;
	} else if (z <= 5.5e+15) {
		tmp = fma((b * t), i, (fma(-b, z, (j * a)) * c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-x), a, Float64(i * b)) * t)
	t_2 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -6.8e+34)
		tmp = t_2;
	elseif (z <= -2.2e-193)
		tmp = t_1;
	elseif (z <= 1.3e-297)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (z <= 3.8e-195)
		tmp = t_1;
	elseif (z <= 5.5e+15)
		tmp = fma(Float64(b * t), i, Float64(fma(Float64(-b), z, Float64(j * a)) * c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.8e+34], t$95$2, If[LessEqual[z, -2.2e-193], t$95$1, If[LessEqual[z, 1.3e-297], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 3.8e-195], t$95$1, If[LessEqual[z, 5.5e+15], N[(N[(b * t), $MachinePrecision] * i + N[(N[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.7999999999999999e34 or 5.5e15 < z

   1. Initial program 73.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -6.7999999999999999e34 < z < -2.19999999999999977e-193 or 1.3e-297 < z < 3.80000000000000013e-195

   1. Initial program 69.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if -2.19999999999999977e-193 < z < 1.3e-297

   1. Initial program 78.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 3.80000000000000013e-195 < z < 5.5e15

   1. Initial program 70.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t, i, \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -3 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -3e+38)
     t_1
     (if (<= z 6e+15)
       (fma (fma (- c) z (* i t)) b (* (fma (- x) t (* j c)) a))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -3e+38) {
		tmp = t_1;
	} else if (z <= 6e+15) {
		tmp = fma(fma(-c, z, (i * t)), b, (fma(-x, t, (j * c)) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -3e+38)
		tmp = t_1;
	elseif (z <= 6e+15)
		tmp = fma(fma(Float64(-c), z, Float64(i * t)), b, Float64(fma(Float64(-x), t, Float64(j * c)) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3e+38], t$95$1, If[LessEqual[z, 6e+15], N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-x) * t + N[(j * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000001e38 or 6e15 < z

   1. Initial program 73.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 68.6

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

   5. Applied rewrites 68.6%

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

   if -3.0000000000000001e38 < z < 6e15

   1. Initial program 71.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 71.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \mathsf{fma}\left(-x, t, j \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ t_2 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 16500000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- x) a (* i b)) t)) (t_2 (* (fma (- c) b (* y x)) z)))
   (if (<= z -6.8e+34)
     t_2
     (if (<= z -2.2e-193)
       t_1
       (if (<= z 1.3e-297)
         (* (fma (- i) y (* c a)) j)
         (if (<= z 16500000000.0) 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(-x, a, (i * b)) * t;
	double t_2 = fma(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -6.8e+34) {
		tmp = t_2;
	} else if (z <= -2.2e-193) {
		tmp = t_1;
	} else if (z <= 1.3e-297) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (z <= 16500000000.0) {
		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(-x), a, Float64(i * b)) * t)
	t_2 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -6.8e+34)
		tmp = t_2;
	elseif (z <= -2.2e-193)
		tmp = t_1;
	elseif (z <= 1.3e-297)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (z <= 16500000000.0)
		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[((-x) * a + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.8e+34], t$95$2, If[LessEqual[z, -2.2e-193], t$95$1, If[LessEqual[z, 1.3e-297], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 16500000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.7999999999999999e34 or 1.65e10 < z

   1. Initial program 73.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -6.7999999999999999e34 < z < -2.19999999999999977e-193 or 1.3e-297 < z < 1.65e10

   1. Initial program 70.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -2.19999999999999977e-193 < z < 1.3e-297

   1. Initial program 78.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 16500000000:\\ \;\;\;\;\mathsf{fma}\left(-x, a, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -3.4e+75)
     t_1
     (if (<= x 1.4e+73) (fma (fma (- c) z (* i t)) b (* (* j c) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -3.4e+75) {
		tmp = t_1;
	} else if (x <= 1.4e+73) {
		tmp = fma(fma(-c, z, (i * t)), b, ((j * c) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -3.4e+75)
		tmp = t_1;
	elseif (x <= 1.4e+73)
		tmp = fma(fma(Float64(-c), z, Float64(i * t)), b, Float64(Float64(j * c) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.4e+75], t$95$1, If[LessEqual[x, 1.4e+73], N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000011e75 or 1.40000000000000004e73 < x

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -3.40000000000000011e75 < x < 1.40000000000000004e73

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 63.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-c, z, i \cdot t\right), b, \left(j \cdot c\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a, j, \mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -3.4e+75)
     t_1
     (if (<= x 1.35e+74) (fma (* c a) j (* (fma (- c) z (* i t)) b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -3.4e+75) {
		tmp = t_1;
	} else if (x <= 1.35e+74) {
		tmp = fma((c * a), j, (fma(-c, z, (i * t)) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -3.4e+75)
		tmp = t_1;
	elseif (x <= 1.35e+74)
		tmp = fma(Float64(c * a), j, Float64(fma(Float64(-c), z, Float64(i * t)) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.4e+75], t$95$1, If[LessEqual[x, 1.35e+74], N[(N[(c * a), $MachinePrecision] * j + N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000011e75 or 1.3499999999999999e74 < x

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -3.40000000000000011e75 < x < 1.3499999999999999e74

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-in N/A

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

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

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

      11. neg-mul-1 N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 62.2%

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

Alternative 7: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- c) b (* y x)) z)))
   (if (<= z -7.5e+34)
     t_1
     (if (<= z 2.2e-143)
       (* (fma (- i) y (* c a)) j)
       (if (<= z 3.3e+15) (* (fma (- c) z (* i t)) b) 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(-c, b, (y * x)) * z;
	double tmp;
	if (z <= -7.5e+34) {
		tmp = t_1;
	} else if (z <= 2.2e-143) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (z <= 3.3e+15) {
		tmp = fma(-c, z, (i * t)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-c), b, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -7.5e+34)
		tmp = t_1;
	elseif (z <= 2.2e-143)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (z <= 3.3e+15)
		tmp = Float64(fma(Float64(-c), z, Float64(i * t)) * b);
	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[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -7.5e+34], t$95$1, If[LessEqual[z, 2.2e-143], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, 3.3e+15], N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.49999999999999976e34 or 3.3e15 < z

   1. Initial program 73.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if -7.49999999999999976e34 < z < 2.19999999999999989e-143

   1. Initial program 72.7%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if 2.19999999999999989e-143 < z < 3.3e15

   1. Initial program 68.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -3.15 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -3.15e+75)
     t_1
     (if (<= x -5.3e-232)
       (* (fma (- b) z (* j a)) c)
       (if (<= x 3.3e-32) (* (fma (- c) z (* i t)) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -3.15e+75) {
		tmp = t_1;
	} else if (x <= -5.3e-232) {
		tmp = fma(-b, z, (j * a)) * c;
	} else if (x <= 3.3e-32) {
		tmp = fma(-c, z, (i * t)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -3.15e+75)
		tmp = t_1;
	elseif (x <= -5.3e-232)
		tmp = Float64(fma(Float64(-b), z, Float64(j * a)) * c);
	elseif (x <= 3.3e-32)
		tmp = Float64(fma(Float64(-c), z, Float64(i * t)) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.15e+75], t$95$1, If[LessEqual[x, -5.3e-232], N[(N[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 3.3e-32], N[(N[((-c) * z + N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.15000000000000018e75 or 3.30000000000000025e-32 < x

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -3.15000000000000018e75 < x < -5.3000000000000003e-232

   1. Initial program 75.3%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

   if -5.3000000000000003e-232 < x < 3.30000000000000025e-32

   1. Initial program 72.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(-c, z, i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{if}\;c \leq -1.05 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) z (* j a)) c)))
   (if (<= c -1.05e+84)
     t_1
     (if (<= c -1.9e-169)
       (* (fma (- c) b (* y x)) z)
       (if (<= c 6.5e+34) (* (fma (- a) t (* 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(-b, z, (j * a)) * c;
	double tmp;
	if (c <= -1.05e+84) {
		tmp = t_1;
	} else if (c <= -1.9e-169) {
		tmp = fma(-c, b, (y * x)) * z;
	} else if (c <= 6.5e+34) {
		tmp = fma(-a, t, (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(-b), z, Float64(j * a)) * c)
	tmp = 0.0
	if (c <= -1.05e+84)
		tmp = t_1;
	elseif (c <= -1.9e-169)
		tmp = Float64(fma(Float64(-c), b, Float64(y * x)) * z);
	elseif (c <= 6.5e+34)
		tmp = Float64(fma(Float64(-a), t, 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[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -1.05e+84], t$95$1, If[LessEqual[c, -1.9e-169], N[(N[((-c) * b + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[c, 6.5e+34], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.05000000000000009e84 or 6.50000000000000017e34 < c

   1. Initial program 63.3%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -1.05000000000000009e84 < c < -1.9e-169

   1. Initial program 77.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if -1.9e-169 < c < 6.50000000000000017e34

   1. Initial program 81.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(-c, b, y \cdot x\right) \cdot z\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.15 \cdot 10^{-232}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-162}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (<= x -8.2e-86)
     t_1
     (if (<= x -3.15e-232)
       (* (* j c) a)
       (if (<= x 4.3e-162) (* (* i t) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, t, (z * y)) * x;
	double tmp;
	if (x <= -8.2e-86) {
		tmp = t_1;
	} else if (x <= -3.15e-232) {
		tmp = (j * c) * a;
	} else if (x <= 4.3e-162) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -8.2e-86)
		tmp = t_1;
	elseif (x <= -3.15e-232)
		tmp = Float64(Float64(j * c) * a);
	elseif (x <= 4.3e-162)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -8.2e-86], t$95$1, If[LessEqual[x, -3.15e-232], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 4.3e-162], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.19999999999999959e-86 or 4.29999999999999996e-162 < x

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

   if -8.19999999999999959e-86 < x < -3.15000000000000005e-232

   1. Initial program 69.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 40.5%

      \[\leadsto \left(c \cdot j\right) \cdot a \]

   if -3.15000000000000005e-232 < x < 4.29999999999999996e-162

   1. Initial program 70.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 42.6%

      \[\leadsto \left(t \cdot i\right) \cdot b \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -3.15 \cdot 10^{-232}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-162}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-183}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-296}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -3.4e+37)
     t_1
     (if (<= z -7.5e-183)
       (* (* (- x) t) a)
       (if (<= z 7e-296) (* (* j c) a) (if (<= z 4e+17) (* (* i t) b) 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 = (z * y) * x;
	double tmp;
	if (z <= -3.4e+37) {
		tmp = t_1;
	} else if (z <= -7.5e-183) {
		tmp = (-x * t) * a;
	} else if (z <= 7e-296) {
		tmp = (j * c) * a;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} 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 = (z * y) * x
    if (z <= (-3.4d+37)) then
        tmp = t_1
    else if (z <= (-7.5d-183)) then
        tmp = (-x * t) * a
    else if (z <= 7d-296) then
        tmp = (j * c) * a
    else if (z <= 4d+17) then
        tmp = (i * t) * b
    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 = (z * y) * x;
	double tmp;
	if (z <= -3.4e+37) {
		tmp = t_1;
	} else if (z <= -7.5e-183) {
		tmp = (-x * t) * a;
	} else if (z <= 7e-296) {
		tmp = (j * c) * a;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -3.4e+37:
		tmp = t_1
	elif z <= -7.5e-183:
		tmp = (-x * t) * a
	elif z <= 7e-296:
		tmp = (j * c) * a
	elif z <= 4e+17:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -3.4e+37)
		tmp = t_1;
	elseif (z <= -7.5e-183)
		tmp = Float64(Float64(Float64(-x) * t) * a);
	elseif (z <= 7e-296)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 4e+17)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -3.4e+37)
		tmp = t_1;
	elseif (z <= -7.5e-183)
		tmp = (-x * t) * a;
	elseif (z <= 7e-296)
		tmp = (j * c) * a;
	elseif (z <= 4e+17)
		tmp = (i * t) * b;
	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[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -3.4e+37], t$95$1, If[LessEqual[z, -7.5e-183], N[(N[((-x) * t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 7e-296], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 4e+17], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.40000000000000006e37 or 4e17 < z

   1. Initial program 73.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 47.4%

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

   if -3.40000000000000006e37 < z < -7.5000000000000004e-183

   1. Initial program 72.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 39.9%

      \[\leadsto \left(\left(-x\right) \cdot t\right) \cdot a \]

   if -7.5000000000000004e-183 < z < 6.9999999999999998e-296

   1. Initial program 76.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 45.1%

      \[\leadsto \left(c \cdot j\right) \cdot a \]

   if 6.9999999999999998e-296 < z < 4e17

   1. Initial program 70.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 40.5%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-183}:\\ \;\;\;\;\left(\left(-x\right) \cdot t\right) \cdot a\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-296}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-296}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -3.4e+37)
     t_1
     (if (<= z -5.8e-182)
       (* (* (- a) t) x)
       (if (<= z 7e-296) (* (* j c) a) (if (<= z 4e+17) (* (* i t) b) 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 = (z * y) * x;
	double tmp;
	if (z <= -3.4e+37) {
		tmp = t_1;
	} else if (z <= -5.8e-182) {
		tmp = (-a * t) * x;
	} else if (z <= 7e-296) {
		tmp = (j * c) * a;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} 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 = (z * y) * x
    if (z <= (-3.4d+37)) then
        tmp = t_1
    else if (z <= (-5.8d-182)) then
        tmp = (-a * t) * x
    else if (z <= 7d-296) then
        tmp = (j * c) * a
    else if (z <= 4d+17) then
        tmp = (i * t) * b
    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 = (z * y) * x;
	double tmp;
	if (z <= -3.4e+37) {
		tmp = t_1;
	} else if (z <= -5.8e-182) {
		tmp = (-a * t) * x;
	} else if (z <= 7e-296) {
		tmp = (j * c) * a;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -3.4e+37:
		tmp = t_1
	elif z <= -5.8e-182:
		tmp = (-a * t) * x
	elif z <= 7e-296:
		tmp = (j * c) * a
	elif z <= 4e+17:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -3.4e+37)
		tmp = t_1;
	elseif (z <= -5.8e-182)
		tmp = Float64(Float64(Float64(-a) * t) * x);
	elseif (z <= 7e-296)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 4e+17)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -3.4e+37)
		tmp = t_1;
	elseif (z <= -5.8e-182)
		tmp = (-a * t) * x;
	elseif (z <= 7e-296)
		tmp = (j * c) * a;
	elseif (z <= 4e+17)
		tmp = (i * t) * b;
	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[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -3.4e+37], t$95$1, If[LessEqual[z, -5.8e-182], N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 7e-296], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 4e+17], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.40000000000000006e37 or 4e17 < z

   1. Initial program 73.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 47.4%

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

   if -3.40000000000000006e37 < z < -5.79999999999999974e-182

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 39.9%

      \[\leadsto \left(\left(-t\right) \cdot a\right) \cdot x \]

   if -5.79999999999999974e-182 < z < 6.9999999999999998e-296

   1. Initial program 76.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 45.1%

      \[\leadsto \left(c \cdot j\right) \cdot a \]

   if 6.9999999999999998e-296 < z < 4e17

   1. Initial program 70.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 40.5%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-296}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) z (* j a)) c)))
   (if (<= c -2.15e-52)
     t_1
     (if (<= c 6.5e+34) (* (fma (- a) t (* 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(-b, z, (j * a)) * c;
	double tmp;
	if (c <= -2.15e-52) {
		tmp = t_1;
	} else if (c <= 6.5e+34) {
		tmp = fma(-a, t, (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(-b), z, Float64(j * a)) * c)
	tmp = 0.0
	if (c <= -2.15e-52)
		tmp = t_1;
	elseif (c <= 6.5e+34)
		tmp = Float64(fma(Float64(-a), t, 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[((-b) * z + N[(j * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -2.15e-52], t$95$1, If[LessEqual[c, 6.5e+34], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.1500000000000002e-52 or 6.50000000000000017e34 < c

   1. Initial program 65.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if -2.1500000000000002e-52 < c < 6.50000000000000017e34

   1. Initial program 81.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.15 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot a\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-296}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -1.15e-31)
     t_1
     (if (<= z 7e-296) (* (* j c) a) (if (<= z 4e+17) (* (* i t) b) 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 = (z * y) * x;
	double tmp;
	if (z <= -1.15e-31) {
		tmp = t_1;
	} else if (z <= 7e-296) {
		tmp = (j * c) * a;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} 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 = (z * y) * x
    if (z <= (-1.15d-31)) then
        tmp = t_1
    else if (z <= 7d-296) then
        tmp = (j * c) * a
    else if (z <= 4d+17) then
        tmp = (i * t) * b
    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 = (z * y) * x;
	double tmp;
	if (z <= -1.15e-31) {
		tmp = t_1;
	} else if (z <= 7e-296) {
		tmp = (j * c) * a;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -1.15e-31:
		tmp = t_1
	elif z <= 7e-296:
		tmp = (j * c) * a
	elif z <= 4e+17:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -1.15e-31)
		tmp = t_1;
	elseif (z <= 7e-296)
		tmp = Float64(Float64(j * c) * a);
	elseif (z <= 4e+17)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -1.15e-31)
		tmp = t_1;
	elseif (z <= 7e-296)
		tmp = (j * c) * a;
	elseif (z <= 4e+17)
		tmp = (i * t) * b;
	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[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.15e-31], t$95$1, If[LessEqual[z, 7e-296], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 4e+17], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1499999999999999e-31 or 4e17 < z

   1. Initial program 73.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 45.0%

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

   if -1.1499999999999999e-31 < z < 6.9999999999999998e-296

   1. Initial program 73.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in c around inf

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 38.3%

      \[\leadsto \left(c \cdot j\right) \cdot a \]

   if 6.9999999999999998e-296 < z < 4e17

   1. Initial program 70.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 40.5%

      \[\leadsto \left(t \cdot i\right) \cdot b \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-31}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-296}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= z -2.5e-10) t_1 (if (<= z 4e+17) (* (* i t) b) 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 = (z * y) * x;
	double tmp;
	if (z <= -2.5e-10) {
		tmp = t_1;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} 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 = (z * y) * x
    if (z <= (-2.5d-10)) then
        tmp = t_1
    else if (z <= 4d+17) then
        tmp = (i * t) * b
    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 = (z * y) * x;
	double tmp;
	if (z <= -2.5e-10) {
		tmp = t_1;
	} else if (z <= 4e+17) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if z <= -2.5e-10:
		tmp = t_1
	elif z <= 4e+17:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (z <= -2.5e-10)
		tmp = t_1;
	elseif (z <= 4e+17)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (z <= -2.5e-10)
		tmp = t_1;
	elseif (z <= 4e+17)
		tmp = (i * t) * b;
	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[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -2.5e-10], t$95$1, If[LessEqual[z, 4e+17], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000016e-10 or 4e17 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.9%

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

   if -2.50000000000000016e-10 < z < 4e17

   1. Initial program 71.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. neg-mul-1 N/A

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

      17. lower-neg.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 32.2%

      \[\leadsto \left(t \cdot i\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-10}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 22.2% 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 72.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
4. 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. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 43.5

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

5. Applied rewrites 43.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 27.7%

   \[\leadsto \left(z \cdot y\right) \cdot x \]
9. Add Preprocessing

Developer Target 1: 60.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

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