Data.Colour.Matrix:inverse from colour-2.3.3, B

Percentage Accurate: 91.6% → 97.2%

Time: 12.6s

Alternatives: 9

Speedup: 0.7×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) - (z * t)) / a
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - (z * t)) / a

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - (z * t)) / a;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) - (z * t)) / a
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - (z * t)) / a

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - (z * t)) / a;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \left(-t\right) \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, x, -z\right)}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (fma (/ y a) x (* (- t) (/ z a)))
     (if (<= t_1 2e+305) (/ t_1 a) (* (/ (fma (/ y t) x (- z)) a) t)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y / a), x, (-t * (z / a)));
	} else if (t_1 <= 2e+305) {
		tmp = t_1 / a;
	} else {
		tmp = (fma((y / t), x, -z) / a) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y / a), x, Float64(Float64(-t) * Float64(z / a)));
	elseif (t_1 <= 2e+305)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(fma(Float64(y / t), x, Float64(-z)) / a) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / a), $MachinePrecision] * x + N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[(t$95$1 / a), $MachinePrecision], N[(N[(N[(N[(y / t), $MachinePrecision] * x + (-z)), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 67.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e305

   1. Initial program 99.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1.9999999999999999e305 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 61.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 99.7%

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

Alternative 2: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+284}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-t}{x}, z, y\right)}{a} \cdot x\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, x, -z\right)}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* z t)) a)))
   (if (<= t_1 -1e+284)
     (* (/ (fma (/ (- t) x) z y) a) x)
     (if (<= t_1 1e+284) t_1 (* (/ (fma (/ y t) x (- z)) a) t)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= -1e+284) {
		tmp = (fma((-t / x), z, y) / a) * x;
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = (fma((y / t), x, -z) / a) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_1 <= -1e+284)
		tmp = Float64(Float64(fma(Float64(Float64(-t) / x), z, y) / a) * x);
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = Float64(Float64(fma(Float64(y / t), x, Float64(-z)) / a) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+284], N[(N[(N[(N[((-t) / x), $MachinePrecision] * z + y), $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], t$95$1, N[(N[(N[(N[(y / t), $MachinePrecision] * x + (-z)), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -1.00000000000000008e284

   1. Initial program 76.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 88.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.3

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

   10. Applied rewrites 88.3%

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

   if -1.00000000000000008e284 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.00000000000000008e284

   1. Initial program 98.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1.00000000000000008e284 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 81.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 94.8%

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

Alternative 3: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot \frac{x}{z} - t}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, x, -z\right)}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* z t)) a)))
   (if (<= t_1 (- INFINITY))
     (* (/ (- (* y (/ x z)) t) a) z)
     (if (<= t_1 1e+284) t_1 (* (/ (fma (/ y t) x (- z)) a) t)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((y * (x / z)) - t) / a) * z;
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = (fma((y / t), x, -z) / a) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(y * Float64(x / z)) - t) / a) * z);
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = Float64(Float64(fma(Float64(y / t), x, Float64(-z)) / a) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], t$95$1, N[(N[(N[(N[(y / t), $MachinePrecision] * x + (-z)), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -inf.0

   1. Initial program 74.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z} - t}{a} \cdot z} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.3%

      \[\leadsto \frac{y \cdot \frac{x}{z} - t}{a} \cdot z \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.00000000000000008e284

   1. Initial program 99.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1.00000000000000008e284 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 81.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 94.8%

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

Alternative 4: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;\frac{y \cdot \frac{x}{z} - t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+305)))
     (* (/ (- (* y (/ x z)) t) a) z)
     (/ t_1 a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+305)) {
		tmp = (((y * (x / z)) - t) / a) * z;
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+305)) {
		tmp = (((y * (x / z)) - t) / a) * z;
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+305):
		tmp = (((y * (x / z)) - t) / a) * z
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+305))
		tmp = Float64(Float64(Float64(Float64(y * Float64(x / z)) - t) / a) * z);
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+305)))
		tmp = (((y * (x / z)) - t) / a) * z;
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+305]], $MachinePrecision]], N[(N[(N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1.9999999999999999e305 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 65.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-neg-in N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z} - t}{a} \cdot z} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.0%

      \[\leadsto \frac{y \cdot \frac{x}{z} - t}{a} \cdot z \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e305

   1. Initial program 99.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;\frac{y \cdot \frac{x}{z} - t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2000:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -4e-52)
   (* (/ y a) x)
   (if (<= (* x y) 2000.0) (* (/ (- z) a) t) (/ (* x y) a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -4e-52) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 2000.0) {
		tmp = (-z / a) * t;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((x * y) <= (-4d-52)) then
        tmp = (y / a) * x
    else if ((x * y) <= 2000.0d0) then
        tmp = (-z / a) * t
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -4e-52) {
		tmp = (y / a) * x;
	} else if ((x * y) <= 2000.0) {
		tmp = (-z / a) * t;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -4e-52:
		tmp = (y / a) * x
	elif (x * y) <= 2000.0:
		tmp = (-z / a) * t
	else:
		tmp = (x * y) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -4e-52)
		tmp = Float64(Float64(y / a) * x);
	elseif (Float64(x * y) <= 2000.0)
		tmp = Float64(Float64(Float64(-z) / a) * t);
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -4e-52)
		tmp = (y / a) * x;
	elseif ((x * y) <= 2000.0)
		tmp = (-z / a) * t;
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e-52], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2000.0], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4e-52

   1. Initial program 87.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      3. lower-/.f64 71.8

         \[\leadsto \color{blue}{\frac{x}{a}} \cdot y \]

   5. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.9%

      \[\leadsto \frac{y}{a} \cdot \color{blue}{x} \]

   if -4e-52 < (*.f64 x y) < 2e3

   1. Initial program 94.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot z}{a}} \cdot t \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot z}{a}} \cdot t \]

      7. mul-1-neg N/A

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

      8. lower-neg.f64 80.5

         \[\leadsto \frac{\color{blue}{-z}}{a} \cdot t \]

   7. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]

   if 2e3 < (*.f64 x y)

   1. Initial program 93.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
   7. Step-by-step derivation

      1. lower-*.f64 80.7

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]

   8. Applied rewrites 80.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2000:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (* (/ (- z) a) t) (/ (- (* x y) (* z t)) a)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-z / a) * t;
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-z / a) * t;
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-z / a) * t
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-z) / a) * t);
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-z / a) * t;
	else
		tmp = ((x * y) - (z * t)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 47.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot z}{a}} \cdot t \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot z}{a}} \cdot t \]

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.6

         \[\leadsto \frac{\color{blue}{-z}}{a} \cdot t \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]

   if -inf.0 < (*.f64 z t)

   1. Initial program 95.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* z t)) -1e+167) (* (/ x a) y) (/ (* x y) a)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - (z * t)) <= -1e+167) {
		tmp = (x / a) * y;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (((x * y) - (z * t)) <= (-1d+167)) then
        tmp = (x / a) * y
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - (z * t)) <= -1e+167) {
		tmp = (x / a) * y;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((x * y) - (z * t)) <= -1e+167:
		tmp = (x / a) * y
	else:
		tmp = (x * y) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(z * t)) <= -1e+167)
		tmp = Float64(Float64(x / a) * y);
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((x * y) - (z * t)) <= -1e+167)
		tmp = (x / a) * y;
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision], -1e+167], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1e167

   1. Initial program 85.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      3. lower-/.f64 42.0

         \[\leadsto \color{blue}{\frac{x}{a}} \cdot y \]

   5. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   if -1e167 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 94.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
   7. Step-by-step derivation

      1. lower-*.f64 57.9

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]

   8. Applied rewrites 57.9%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 6.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a 6.6e-110) (* (/ y a) x) (* (/ x a) y)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 6.6e-110) {
		tmp = (y / a) * x;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (a <= 6.6d-110) then
        tmp = (y / a) * x
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 6.6e-110) {
		tmp = (y / a) * x;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if a <= 6.6e-110:
		tmp = (y / a) * x
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 6.6e-110)
		tmp = Float64(Float64(y / a) * x);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 6.6e-110)
		tmp = (y / a) * x;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, 6.6e-110], N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.5999999999999998e-110

   1. Initial program 94.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      3. lower-/.f64 49.3

         \[\leadsto \color{blue}{\frac{x}{a}} \cdot y \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.9%

      \[\leadsto \frac{y}{a} \cdot \color{blue}{x} \]

   if 6.5999999999999998e-110 < a

   1. Initial program 87.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

      3. lower-/.f64 50.7

         \[\leadsto \color{blue}{\frac{x}{a}} \cdot y \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{x}{a} \cdot y \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* (/ x a) y))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (x / a) * y;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    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
    code = (x / a) * y
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return (x / a) * y;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (x / a) * y

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(x / a) * y)
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (x / a) * y;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{x \cdot y - z \cdot t}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   3. lower-/.f64 49.8

      \[\leadsto \color{blue}{\frac{x}{a}} \cdot y \]

5. Applied rewrites 49.8%

   \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

6. Final simplification 49.8%

   \[\leadsto \frac{x}{a} \cdot y \]
7. Add Preprocessing

Developer Target 1: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

  (/ (- (* x y) (* z t)) a))