Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.4% → 82.4%

Time: 16.5s

Alternatives: 18

Speedup: 0.9×

• 57.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 60.4

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

   5. Simplified 60.4%

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

4. Final simplification 87.1%

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

Alternative 2: 64.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -4.3 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -3.6 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right)\right)\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(x, \frac{y \cdot z}{c}, \mathsf{fma}\left(-i, \frac{y \cdot j}{c}, t \cdot j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* a b)))))
   (if (<= i -4.3e+92)
     t_1
     (if (<= i -3.6e-69)
       (fma (- (* t c) (* y i)) j (* x (* y z)))
       (if (<= i 5.5e+59)
         (fma z (fma b (- c) (* x y)) (* t (fma c j (* x (- a)))))
         (if (<= i 8.2e+180)
           t_1
           (* c (fma x (/ (* y z) c) (fma (- i) (/ (* y j) c) (* t j))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -4.3e+92) {
		tmp = t_1;
	} else if (i <= -3.6e-69) {
		tmp = fma(((t * c) - (y * i)), j, (x * (y * z)));
	} else if (i <= 5.5e+59) {
		tmp = fma(z, fma(b, -c, (x * y)), (t * fma(c, j, (x * -a))));
	} else if (i <= 8.2e+180) {
		tmp = t_1;
	} else {
		tmp = c * fma(x, ((y * z) / c), fma(-i, ((y * j) / c), (t * j)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -4.3e+92)
		tmp = t_1;
	elseif (i <= -3.6e-69)
		tmp = fma(Float64(Float64(t * c) - Float64(y * i)), j, Float64(x * Float64(y * z)));
	elseif (i <= 5.5e+59)
		tmp = fma(z, fma(b, Float64(-c), Float64(x * y)), Float64(t * fma(c, j, Float64(x * Float64(-a)))));
	elseif (i <= 8.2e+180)
		tmp = t_1;
	else
		tmp = Float64(c * fma(x, Float64(Float64(y * z) / c), fma(Float64(-i), Float64(Float64(y * j) / c), Float64(t * j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -4.3e+92], t$95$1, If[LessEqual[i, -3.6e-69], N[(N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.5e+59], N[(z * N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t * N[(c * j + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.2e+180], t$95$1, N[(c * N[(x * N[(N[(y * z), $MachinePrecision] / c), $MachinePrecision] + N[((-i) * N[(N[(y * j), $MachinePrecision] / c), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -4.2999999999999998e92 or 5.4999999999999999e59 < i < 8.2e180

   1. Initial program 65.4%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.6

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

   5. Simplified 71.6%

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

   if -4.2999999999999998e92 < i < -3.60000000000000018e-69

   1. Initial program 74.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.6

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

   5. Simplified 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 75.3

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 72.5

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

   7. Applied egg-rr 72.5%

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

   if -3.60000000000000018e-69 < i < 5.4999999999999999e59

   1. Initial program 83.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 85.3%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. mul-1-neg N/A

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

   8. Simplified 76.9%

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

   if 8.2e180 < i

   1. Initial program 73.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.6

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

   5. Simplified 51.6%

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

   6. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 61.7

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

   8. Simplified 61.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.3 \cdot 10^{+92}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -3.6 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right)\right)\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+180}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(x, \frac{y \cdot z}{c}, \mathsf{fma}\left(-i, \frac{y \cdot j}{c}, t \cdot j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - t \cdot a\\ \mathbf{if}\;x \leq -1.46 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_1, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* y z) (* t a))))
   (if (<= x -1.46e+227)
     (fma x t_1 (* j (* t c)))
     (if (<= x 7.2e+256)
       (fma
        t
        (fma j c (* x (- a)))
        (fma y (fma j (- i) (* x z)) (* b (fma c (- z) (* a i)))))
       (* 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 = (y * z) - (t * a);
	double tmp;
	if (x <= -1.46e+227) {
		tmp = fma(x, t_1, (j * (t * c)));
	} else if (x <= 7.2e+256) {
		tmp = fma(t, fma(j, c, (x * -a)), fma(y, fma(j, -i, (x * z)), (b * fma(c, -z, (a * i)))));
	} else {
		tmp = x * t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(y * z) - Float64(t * a))
	tmp = 0.0
	if (x <= -1.46e+227)
		tmp = fma(x, t_1, Float64(j * Float64(t * c)));
	elseif (x <= 7.2e+256)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), fma(y, fma(j, Float64(-i), Float64(x * z)), Float64(b * fma(c, Float64(-z), Float64(a * i)))));
	else
		tmp = Float64(x * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.46e+227], N[(x * t$95$1 + N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e+256], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(y * N[(j * (-i) + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45999999999999993e227

   1. Initial program 77.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 83.3

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

   5. Simplified 83.3%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 88.9

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

   8. Simplified 88.9%

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

   if -1.45999999999999993e227 < x < 7.19999999999999942e256

   1. Initial program 76.4%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 85.2%

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

   if 7.19999999999999942e256 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), \mathsf{fma}\left(y, \mathsf{fma}\left(j, -i, x \cdot z\right), b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), i \cdot \left(y \cdot \left(-j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))))
   (if (<= b -1.15e+132)
     t_1
     (if (<= b -7.5e-149)
       (+ (* j (- (* t c) (* y i))) (* y (* x z)))
       (if (<= b 2.3e-160)
         (fma x (- (* y z) (* t a)) (* j (* t c)))
         (if (<= b 2.65e+79)
           (fma t (fma j c (* x (- a))) (* i (* y (- j))))
           t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double tmp;
	if (b <= -1.15e+132) {
		tmp = t_1;
	} else if (b <= -7.5e-149) {
		tmp = (j * ((t * c) - (y * i))) + (y * (x * z));
	} else if (b <= 2.3e-160) {
		tmp = fma(x, ((y * z) - (t * a)), (j * (t * c)));
	} else if (b <= 2.65e+79) {
		tmp = fma(t, fma(j, c, (x * -a)), (i * (y * -j)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	tmp = 0.0
	if (b <= -1.15e+132)
		tmp = t_1;
	elseif (b <= -7.5e-149)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(y * Float64(x * z)));
	elseif (b <= 2.3e-160)
		tmp = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(j * Float64(t * c)));
	elseif (b <= 2.65e+79)
		tmp = fma(t, fma(j, c, Float64(x * Float64(-a))), Float64(i * Float64(y * Float64(-j))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.15e+132], t$95$1, If[LessEqual[b, -7.5e-149], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-160], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.65e+79], N[(t * N[(j * c + N[(x * (-a)), $MachinePrecision]), $MachinePrecision] + N[(i * N[(y * (-j)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.1500000000000001e132 or 2.64999999999999989e79 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 66.6

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

   5. Simplified 66.6%

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

   if -1.1500000000000001e132 < b < -7.49999999999999995e-149

   1. Initial program 77.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.0

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

   5. Simplified 67.0%

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

   if -7.49999999999999995e-149 < b < 2.29999999999999985e-160

   1. Initial program 82.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 82.2

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

   5. Simplified 82.2%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 74.6

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

   8. Simplified 74.6%

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

   if 2.29999999999999985e-160 < b < 2.64999999999999989e79

   1. Initial program 69.1%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 77.2%

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

   6. Taylor expanded in j around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 63.4

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

   8. Simplified 63.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+132}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(j, c, x \cdot \left(-a\right)\right), i \cdot \left(y \cdot \left(-j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{if}\;i \leq -4.3 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -3.6 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right)\right)\\ \mathbf{elif}\;i \leq 9 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y, c \cdot \left(t \cdot j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* a b)))))
   (if (<= i -4.3e+92)
     t_1
     (if (<= i -3.6e-69)
       (fma (- (* t c) (* y i)) j (* x (* y z)))
       (if (<= i 5.5e+59)
         (fma z (fma b (- c) (* x y)) (* t (fma c j (* x (- a)))))
         (if (<= i 9e+180) t_1 (fma z (* x y) (* c (* t j)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (a * b));
	double tmp;
	if (i <= -4.3e+92) {
		tmp = t_1;
	} else if (i <= -3.6e-69) {
		tmp = fma(((t * c) - (y * i)), j, (x * (y * z)));
	} else if (i <= 5.5e+59) {
		tmp = fma(z, fma(b, -c, (x * y)), (t * fma(c, j, (x * -a))));
	} else if (i <= 9e+180) {
		tmp = t_1;
	} else {
		tmp = fma(z, (x * y), (c * (t * j)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(a * b)))
	tmp = 0.0
	if (i <= -4.3e+92)
		tmp = t_1;
	elseif (i <= -3.6e-69)
		tmp = fma(Float64(Float64(t * c) - Float64(y * i)), j, Float64(x * Float64(y * z)));
	elseif (i <= 5.5e+59)
		tmp = fma(z, fma(b, Float64(-c), Float64(x * y)), Float64(t * fma(c, j, Float64(x * Float64(-a)))));
	elseif (i <= 9e+180)
		tmp = t_1;
	else
		tmp = fma(z, Float64(x * y), Float64(c * Float64(t * j)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -4.3e+92], t$95$1, If[LessEqual[i, -3.6e-69], N[(N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.5e+59], N[(z * N[(b * (-c) + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t * N[(c * j + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 9e+180], t$95$1, N[(z * N[(x * y), $MachinePrecision] + N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -4.2999999999999998e92 or 5.4999999999999999e59 < i < 8.99999999999999962e180

   1. Initial program 65.4%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.6

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

   5. Simplified 71.6%

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

   if -4.2999999999999998e92 < i < -3.60000000000000018e-69

   1. Initial program 74.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.6

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

   5. Simplified 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 75.3

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 72.5

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

   7. Applied egg-rr 72.5%

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

   if -3.60000000000000018e-69 < i < 5.4999999999999999e59

   1. Initial program 83.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 85.3%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. mul-1-neg N/A

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

   8. Simplified 76.9%

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

   if 8.99999999999999962e180 < i

   1. Initial program 73.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.6

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

   5. Simplified 51.6%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 59.8

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

   8. Simplified 59.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.3 \cdot 10^{+92}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{elif}\;i \leq -3.6 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, x \cdot y\right), t \cdot \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right)\right)\\ \mathbf{elif}\;i \leq 9 \cdot 10^{+180}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y, c \cdot \left(t \cdot j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(-y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma c (- z) (* a i)))))
   (if (<= b -1.15e+132)
     t_1
     (if (<= b 2.75e+81)
       (fma x (- (* y z) (* t a)) (* j (fma c t (* i (- y)))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(c, -z, (a * i));
	double tmp;
	if (b <= -1.15e+132) {
		tmp = t_1;
	} else if (b <= 2.75e+81) {
		tmp = fma(x, ((y * z) - (t * a)), (j * fma(c, t, (i * -y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(c, Float64(-z), Float64(a * i)))
	tmp = 0.0
	if (b <= -1.15e+132)
		tmp = t_1;
	elseif (b <= 2.75e+81)
		tmp = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(j * fma(c, t, Float64(i * Float64(-y)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.15e+132], t$95$1, If[LessEqual[b, 2.75e+81], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(j * N[(c * t + N[(i * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1500000000000001e132 or 2.7500000000000002e81 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 66.6

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

   5. Simplified 66.6%

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

   if -1.1500000000000001e132 < b < 2.7500000000000002e81

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 71.8

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

   5. Simplified 71.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+132}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(-y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-256}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma x (- (* y z) (* t a)) (* j (* t c)))))
   (if (<= x -5.7e+143)
     t_1
     (if (<= x -1.9e-256)
       (+ (* j (- (* t c) (* y i))) (* y (* x z)))
       (if (<= x 5400000000000.0) (* c (fma j t (* z (- 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(x, ((y * z) - (t * a)), (j * (t * c)));
	double tmp;
	if (x <= -5.7e+143) {
		tmp = t_1;
	} else if (x <= -1.9e-256) {
		tmp = (j * ((t * c) - (y * i))) + (y * (x * z));
	} else if (x <= 5400000000000.0) {
		tmp = c * fma(j, t, (z * -b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(j * Float64(t * c)))
	tmp = 0.0
	if (x <= -5.7e+143)
		tmp = t_1;
	elseif (x <= -1.9e-256)
		tmp = Float64(Float64(j * Float64(Float64(t * c) - Float64(y * i))) + Float64(y * Float64(x * z)));
	elseif (x <= 5400000000000.0)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-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[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.7e+143], t$95$1, If[LessEqual[x, -1.9e-256], N[(N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5400000000000.0], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.70000000000000022e143 or 5.4e12 < x

   1. Initial program 82.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.7

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

   8. Simplified 76.7%

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

   if -5.70000000000000022e143 < x < -1.89999999999999988e-256

   1. Initial program 73.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.6

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

   5. Simplified 57.6%

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

   if -1.89999999999999988e-256 < x < 5.4e12

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 61.8

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

   5. Simplified 61.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-256}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-256}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma x (- (* y z) (* t a)) (* j (* t c)))))
   (if (<= x -5.7e+143)
     t_1
     (if (<= x -1.9e-256)
       (fma (- (* t c) (* y i)) j (* x (* y z)))
       (if (<= x 5400000000000.0) (* c (fma j t (* z (- 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(x, ((y * z) - (t * a)), (j * (t * c)));
	double tmp;
	if (x <= -5.7e+143) {
		tmp = t_1;
	} else if (x <= -1.9e-256) {
		tmp = fma(((t * c) - (y * i)), j, (x * (y * z)));
	} else if (x <= 5400000000000.0) {
		tmp = c * fma(j, t, (z * -b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(j * Float64(t * c)))
	tmp = 0.0
	if (x <= -5.7e+143)
		tmp = t_1;
	elseif (x <= -1.9e-256)
		tmp = fma(Float64(Float64(t * c) - Float64(y * i)), j, Float64(x * Float64(y * z)));
	elseif (x <= 5400000000000.0)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-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[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.7e+143], t$95$1, If[LessEqual[x, -1.9e-256], N[(N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5400000000000.0], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.70000000000000022e143 or 5.4e12 < x

   1. Initial program 82.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.7

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

   8. Simplified 76.7%

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

   if -5.70000000000000022e143 < x < -1.89999999999999988e-256

   1. Initial program 73.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.6

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

   5. Simplified 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 57.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 57.6

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 55.9

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

   7. Applied egg-rr 55.9%

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

   if -1.89999999999999988e-256 < x < 5.4e12

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 61.8

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

   5. Simplified 61.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-256}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{if}\;x \leq -7000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-256}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma x (- (* y z) (* t a)) (* j (* t c)))))
   (if (<= x -7000000000.0)
     t_1
     (if (<= x -1.9e-256)
       (* j (fma i (- y) (* t c)))
       (if (<= x 5400000000000.0) (* c (fma j t (* z (- 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(x, ((y * z) - (t * a)), (j * (t * c)));
	double tmp;
	if (x <= -7000000000.0) {
		tmp = t_1;
	} else if (x <= -1.9e-256) {
		tmp = j * fma(i, -y, (t * c));
	} else if (x <= 5400000000000.0) {
		tmp = c * fma(j, t, (z * -b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(x, Float64(Float64(y * z) - Float64(t * a)), Float64(j * Float64(t * c)))
	tmp = 0.0
	if (x <= -7000000000.0)
		tmp = t_1;
	elseif (x <= -1.9e-256)
		tmp = Float64(j * fma(i, Float64(-y), Float64(t * c)));
	elseif (x <= 5400000000000.0)
		tmp = Float64(c * fma(j, t, Float64(z * Float64(-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[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7000000000.0], t$95$1, If[LessEqual[x, -1.9e-256], N[(j * N[(i * (-y) + N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5400000000000.0], N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e9 or 5.4e12 < x

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in c around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.8

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

   8. Simplified 69.8%

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

   if -7e9 < x < -1.89999999999999988e-256

   1. Initial program 71.3%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 87.0%

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

   6. Taylor expanded in j around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-*.f64 55.3

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

   8. Simplified 55.3%

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

   if -1.89999999999999988e-256 < x < 5.4e12

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 61.8

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

   5. Simplified 61.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-256}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\\ \mathbf{elif}\;x \leq 5400000000000:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(t \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{if}\;x \leq -1.62 \cdot 10^{+248}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq -3.15 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+79}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma t (- x) (* b i)))))
   (if (<= x -1.62e+248)
     (* x (* y z))
     (if (<= x -3.15e-130)
       t_1
       (if (<= x 3.2e+79) (* b (fma c (- z) (* a i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(t, -x, (b * i));
	double tmp;
	if (x <= -1.62e+248) {
		tmp = x * (y * z);
	} else if (x <= -3.15e-130) {
		tmp = t_1;
	} else if (x <= 3.2e+79) {
		tmp = b * fma(c, -z, (a * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(t, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (x <= -1.62e+248)
		tmp = Float64(x * Float64(y * z));
	elseif (x <= -3.15e-130)
		tmp = t_1;
	elseif (x <= 3.2e+79)
		tmp = Float64(b * fma(c, Float64(-z), Float64(a * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.62e+248], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.15e-130], t$95$1, If[LessEqual[x, 3.2e+79], N[(b * N[(c * (-z) + N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.61999999999999988e248

   1. Initial program 84.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.3

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

   8. Simplified 69.3%

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

   if -1.61999999999999988e248 < x < -3.1499999999999998e-130 or 3.20000000000000003e79 < x

   1. Initial program 76.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 53.3

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

   5. Simplified 53.3%

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

   if -3.1499999999999998e-130 < x < 3.20000000000000003e79

   1. Initial program 75.7%

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

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 52.6

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

   5. Simplified 52.6%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{+248}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq -3.15 \cdot 10^{-130}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+79}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{if}\;c \leq -7.5 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma j t (* z (- b))))))
   (if (<= c -7.5e+170)
     t_1
     (if (<= c 2.5e-5) (* i (fma j (- y) (* a 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 = c * fma(j, t, (z * -b));
	double tmp;
	if (c <= -7.5e+170) {
		tmp = t_1;
	} else if (c <= 2.5e-5) {
		tmp = i * fma(j, -y, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(j, t, Float64(z * Float64(-b))))
	tmp = 0.0
	if (c <= -7.5e+170)
		tmp = t_1;
	elseif (c <= 2.5e-5)
		tmp = Float64(i * fma(j, Float64(-y), Float64(a * 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[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7.5e+170], t$95$1, If[LessEqual[c, 2.5e-5], N[(i * N[(j * (-y) + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.5000000000000002e170 or 2.50000000000000012e-5 < c

   1. Initial program 67.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 72.7

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

   5. Simplified 72.7%

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

   if -7.5000000000000002e170 < c < 2.50000000000000012e-5

   1. Initial program 82.7%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 52.3

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

   5. Simplified 52.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+170}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{if}\;c \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (fma j t (* z (- b))))))
   (if (<= c -2.6e+88)
     t_1
     (if (<= c 1.85e-16) (* a (fma t (- x) (* b i))) 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 = c * fma(j, t, (z * -b));
	double tmp;
	if (c <= -2.6e+88) {
		tmp = t_1;
	} else if (c <= 1.85e-16) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * fma(j, t, Float64(z * Float64(-b))))
	tmp = 0.0
	if (c <= -2.6e+88)
		tmp = t_1;
	elseif (c <= 1.85e-16)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(j * t + N[(z * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.6e+88], t$95$1, If[LessEqual[c, 1.85e-16], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.6000000000000001e88 or 1.85e-16 < c

   1. Initial program 68.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 69.3

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

   5. Simplified 69.3%

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

   if -2.6000000000000001e88 < c < 1.85e-16

   1. Initial program 83.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.2

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

   5. Simplified 50.2%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(j, t, z \cdot \left(-b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-15}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -6.5e+110)
   (* t (* c j))
   (if (<= c 2e-15) (* a (fma t (- x) (* b i))) (* c (* t j)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -6.5e+110) {
		tmp = t * (c * j);
	} else if (c <= 2e-15) {
		tmp = a * fma(t, -x, (b * i));
	} else {
		tmp = c * (t * j);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -6.5e+110)
		tmp = Float64(t * Float64(c * j));
	elseif (c <= 2e-15)
		tmp = Float64(a * fma(t, Float64(-x), Float64(b * i)));
	else
		tmp = Float64(c * Float64(t * j));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -6.5e+110], N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2e-15], N[(a * N[(t * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.4999999999999997e110

   1. Initial program 66.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 67.6

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

   5. Simplified 67.6%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 45.8

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

   8. Simplified 45.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 48.2

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

   10. Applied egg-rr 48.2%

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

   if -6.4999999999999997e110 < c < 2.0000000000000002e-15

   1. Initial program 83.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.3

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

   5. Simplified 50.3%

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

   if 2.0000000000000002e-15 < c

   1. Initial program 68.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.7

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

   5. Simplified 59.7%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 46.6

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

   8. Simplified 46.6%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-15}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, -x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+107}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;c \leq 10^{-16}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -8.5e+107)
   (* t (* c j))
   (if (<= c 1e-16) (* b (* a i)) (* c (* t j)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -8.5e+107) {
		tmp = t * (c * j);
	} else if (c <= 1e-16) {
		tmp = b * (a * i);
	} else {
		tmp = c * (t * j);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (c <= (-8.5d+107)) then
        tmp = t * (c * j)
    else if (c <= 1d-16) then
        tmp = b * (a * i)
    else
        tmp = c * (t * j)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -8.5e+107) {
		tmp = t * (c * j);
	} else if (c <= 1e-16) {
		tmp = b * (a * i);
	} else {
		tmp = c * (t * j);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if c <= -8.5e+107:
		tmp = t * (c * j)
	elif c <= 1e-16:
		tmp = b * (a * i)
	else:
		tmp = c * (t * j)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -8.5e+107)
		tmp = Float64(t * Float64(c * j));
	elseif (c <= 1e-16)
		tmp = Float64(b * Float64(a * i));
	else
		tmp = Float64(c * Float64(t * j));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (c <= -8.5e+107)
		tmp = t * (c * j);
	elseif (c <= 1e-16)
		tmp = b * (a * i);
	else
		tmp = c * (t * j);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -8.5e+107], N[(t * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e-16], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.4999999999999999e107

   1. Initial program 66.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 67.6

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

   5. Simplified 67.6%

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

   6. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 45.8

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

   8. Simplified 45.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 48.2

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

   10. Applied egg-rr 48.2%

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

   if -8.4999999999999999e107 < c < 9.9999999999999998e-17

   1. Initial program 83.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 50.3

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

   5. Simplified 50.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.7

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

   8. Simplified 35.7%

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

   if 9.9999999999999998e-17 < c

   1. Initial program 68.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.7

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

   5. Simplified 59.7%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 46.6

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

   8. Simplified 46.6%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+107}:\\ \;\;\;\;t \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;c \leq 10^{-16}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+107}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;c \leq 10^{-16}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -9e+107)
   (* j (* t c))
   (if (<= c 1e-16) (* b (* a i)) (* c (* t j)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -9e+107) {
		tmp = j * (t * c);
	} else if (c <= 1e-16) {
		tmp = b * (a * i);
	} else {
		tmp = c * (t * j);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (c <= (-9d+107)) then
        tmp = j * (t * c)
    else if (c <= 1d-16) then
        tmp = b * (a * i)
    else
        tmp = c * (t * j)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -9e+107) {
		tmp = j * (t * c);
	} else if (c <= 1e-16) {
		tmp = b * (a * i);
	} else {
		tmp = c * (t * j);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if c <= -9e+107:
		tmp = j * (t * c)
	elif c <= 1e-16:
		tmp = b * (a * i)
	else:
		tmp = c * (t * j)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -9e+107)
		tmp = Float64(j * Float64(t * c));
	elseif (c <= 1e-16)
		tmp = Float64(b * Float64(a * i));
	else
		tmp = Float64(c * Float64(t * j));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (c <= -9e+107)
		tmp = j * (t * c);
	elseif (c <= 1e-16)
		tmp = b * (a * i);
	else
		tmp = c * (t * j);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -9e+107], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e-16], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -9e107

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 74.6%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(b, -1 \cdot c, x \cdot y\right)}, t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(c\right)}, x \cdot y\right), t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(c\right)}, x \cdot y\right), t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), \color{blue}{y \cdot x}\right), t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), \color{blue}{y \cdot x}\right), t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), y \cdot x\right), \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)}\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), y \cdot x\right), t \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(a \cdot x\right)\right)}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), y \cdot x\right), t \cdot \color{blue}{\mathsf{fma}\left(c, j, -1 \cdot \left(a \cdot x\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(b, \mathsf{neg}\left(c\right), y \cdot x\right), t \cdot \mathsf{fma}\left(c, j, \color{blue}{\mathsf{neg}\left(a \cdot x\right)}\right)\right) \]

   8. Simplified 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(b, -c, y \cdot x\right), t \cdot \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right)\right)} \]

   9. Taylor expanded in j around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(j \cdot t\right) \cdot c} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{j \cdot \left(t \cdot c\right)} \]

       3. *-commutative N/A

          \[\leadsto j \cdot \color{blue}{\left(c \cdot t\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{j \cdot \left(c \cdot t\right)} \]

       5. *-commutative N/A

          \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

       6. lower-*.f64 46.1

          \[\leadsto j \cdot \color{blue}{\left(t \cdot c\right)} \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{j \cdot \left(t \cdot c\right)} \]

   if -9e107 < c < 9.9999999999999998e-17

   1. Initial program 83.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 50.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

      5. lower-*.f64 35.7

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i\right)} \]

   8. Simplified 35.7%

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

   if 9.9999999999999998e-17 < c

   1. Initial program 68.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-*.f64 59.7

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 59.7%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. lower-*.f64 46.6

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+107}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;c \leq 10^{-16}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;c \leq -9 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 10^{-16}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= c -9e+107) t_1 (if (<= c 1e-16) (* b (* a i)) 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 = c * (t * j);
	double tmp;
	if (c <= -9e+107) {
		tmp = t_1;
	} else if (c <= 1e-16) {
		tmp = b * (a * i);
	} 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 = c * (t * j)
    if (c <= (-9d+107)) then
        tmp = t_1
    else if (c <= 1d-16) then
        tmp = b * (a * i)
    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 = c * (t * j);
	double tmp;
	if (c <= -9e+107) {
		tmp = t_1;
	} else if (c <= 1e-16) {
		tmp = b * (a * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if c <= -9e+107:
		tmp = t_1
	elif c <= 1e-16:
		tmp = b * (a * i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (c <= -9e+107)
		tmp = t_1;
	elseif (c <= 1e-16)
		tmp = Float64(b * Float64(a * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (c <= -9e+107)
		tmp = t_1;
	elseif (c <= 1e-16)
		tmp = b * (a * i);
	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[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9e+107], t$95$1, If[LessEqual[c, 1e-16], N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -9e107 or 9.9999999999999998e-17 < c

   1. Initial program 67.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. lower-*.f64 59.5

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 59.5%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. lower-*.f64 46.3

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 46.3%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

   if -9e107 < c < 9.9999999999999998e-17

   1. Initial program 83.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. lower-*.f64 50.3

          \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

      5. lower-*.f64 35.7

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i\right)} \]

   8. Simplified 35.7%

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+107}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;c \leq 10^{-16}:\\ \;\;\;\;b \cdot \left(a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 22.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(a \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* b (* a i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return b * (a * 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 = b * (a * 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 b * (a * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return b * (a * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(b * Float64(a * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = b * (a * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(b * N[(a * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

   12. lower-*.f64 40.9

       \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

5. Simplified 40.9%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot i} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(b \cdot a\right)} \cdot i \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]

   5. lower-*.f64 26.2

      \[\leadsto b \cdot \color{blue}{\left(a \cdot i\right)} \]

8. Simplified 26.2%

   \[\leadsto \color{blue}{b \cdot \left(a \cdot i\right)} \]
9. Add Preprocessing

Alternative 18: 22.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* b i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * 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 = a * (b * 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 a * (b * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (b * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(b * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (b * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto a \cdot \mathsf{fma}\left(t, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

   12. lower-*.f64 40.9

       \[\leadsto a \cdot \mathsf{fma}\left(t, -x, \color{blue}{i \cdot b}\right) \]

5. Simplified 40.9%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, -x, i \cdot b\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 25.4

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]

8. Simplified 25.4%

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
9. Add Preprocessing

Developer Target 1: 68.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))