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

Percentage Accurate: 91.6% → 96.5%

Time: 10.9s

Alternatives: 11

Speedup: 1.0×

• 27.8% 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: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, \frac{y}{t \cdot a}, \frac{z}{-a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, x \cdot \frac{y}{a}\right)\\ \end{array} \end{array} \]

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 -5e+301)
     (* t (fma x (/ y (* t a)) (/ z (- a))))
     (if (<= t_1 2e+266)
       (/ (fma y x (* z (- t))) a)
       (fma (- z) (/ t a) (* 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t * fma(x, (y / (t * a)), (z / -a));
	} else if (t_1 <= 2e+266) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = fma(-z, (t / a), (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)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -5e+301)
		tmp = Float64(t * fma(x, Float64(y / Float64(t * a)), Float64(z / Float64(-a))));
	elseif (t_1 <= 2e+266)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = fma(Float64(-z), Float64(t / a), Float64(x * Float64(y / a)));
	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, -5e+301], N[(t * N[(x * N[(y / N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(z / (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+266], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-z) * N[(t / a), $MachinePrecision] + N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000004e301

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 85.2

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

   8. Applied rewrites 85.2%

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

   if -5.0000000000000004e301 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000001e266

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if 2.0000000000000001e266 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 79.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 86.7

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

   4. Applied rewrites 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

   6. Applied rewrites 97.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+301}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, \frac{y}{t \cdot a}, \frac{z}{-a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, x \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, x \cdot \frac{y}{a}\right)\\ \end{array} \end{array} \]

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 -5e+280)
     (fma (- z) (/ t a) (* y (/ x a)))
     (if (<= t_1 2e+266)
       (/ (fma y x (* z (- t))) a)
       (fma (- z) (/ t a) (* 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -5e+280) {
		tmp = fma(-z, (t / a), (y * (x / a)));
	} else if (t_1 <= 2e+266) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = fma(-z, (t / a), (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)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -5e+280)
		tmp = fma(Float64(-z), Float64(t / a), Float64(y * Float64(x / a)));
	elseif (t_1 <= 2e+266)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = fma(Float64(-z), Float64(t / a), Float64(x * Float64(y / a)));
	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, -5e+280], N[((-z) * N[(t / a), $MachinePrecision] + N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+266], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-z) * N[(t / a), $MachinePrecision] + N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000002e280

   1. Initial program 76.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -5.0000000000000002e280 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000001e266

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 2.0000000000000001e266 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 79.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 86.7

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

   4. Applied rewrites 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.1

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

   6. Applied rewrites 97.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, x \cdot \frac{y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(-z, \frac{t}{a}, y \cdot \frac{x}{a}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+280}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))) (t_2 (fma (- z) (/ t a) (* y (/ x a)))))
   (if (<= t_1 -5e+280)
     t_2
     (if (<= t_1 1e+308) (/ (fma y x (* z (- t))) a) t_2))))

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 t_2 = fma(-z, (t / a), (y * (x / a)));
	double tmp;
	if (t_1 <= -5e+280) {
		tmp = t_2;
	} else if (t_1 <= 1e+308) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(Float64(-z), Float64(t / a), Float64(y * Float64(x / a)))
	tmp = 0.0
	if (t_1 <= -5e+280)
		tmp = t_2;
	elseif (t_1 <= 1e+308)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[((-z) * N[(t / a), $MachinePrecision] + N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+280], t$95$2, If[LessEqual[t$95$1, 1e+308], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000002e280 or 1e308 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 75.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if -5.0000000000000002e280 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e308

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 97.7

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

   4. Applied rewrites 97.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a}, y \cdot \frac{x}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{a}{t}}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{a}{z}}{t}}\\ \end{array} \end{array} \]

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
 (if (<= (* z t) (- INFINITY))
   (/ -1.0 (/ (/ a t) z))
   (if (<= (* z t) 5e+247)
     (/ (fma y x (* z (- t))) a)
     (/ -1.0 (/ (/ a z) 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 tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = -1.0 / ((a / t) / z);
	} else if ((z * t) <= 5e+247) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = -1.0 / ((a / z) / 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)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(Float64(a / t) / z));
	elseif (Float64(z * t) <= 5e+247)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(-1.0 / Float64(Float64(a / z) / 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_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(-1.0 / N[(N[(a / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+247], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 / N[(N[(a / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 70.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 70.7

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

   7. Applied rewrites 70.7%

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

   9. Applied rewrites 99.7%

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

   if -inf.0 < (*.f64 z t) < 5.00000000000000023e247

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 95.0

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

   4. Applied rewrites 95.0%

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

   if 5.00000000000000023e247 < (*.f64 z t)

   1. Initial program 77.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 78.0

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

   7. Applied rewrites 78.0%

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

   9. Applied rewrites 99.8%

      \[\leadsto \frac{-1}{\frac{\frac{a}{z}}{\color{blue}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{a}{t}}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{a}{z}}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{-1}{\frac{\frac{a}{t}}{z}}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ -1.0 (/ (/ a t) z))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 5e+247) (/ (fma y x (* z (- t))) a) t_1))))

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 = -1.0 / ((a / t) / z);
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 5e+247) {
		tmp = fma(y, x, (z * -t)) / a;
	} else {
		tmp = t_1;
	}
	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(-1.0 / Float64(Float64(a / t) / z))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+247)
		tmp = Float64(fma(y, x, Float64(z * Float64(-t))) / a);
	else
		tmp = t_1;
	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[(-1.0 / N[(N[(a / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+247], N[(N[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 5.00000000000000023e247 < (*.f64 z t)

   1. Initial program 74.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.8

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 z t) < 5.00000000000000023e247

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 95.0

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

   4. Applied rewrites 95.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{a}{t}}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{a}{t}}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z}{a}, \frac{x \cdot y}{a}\right)\\ \end{array} \end{array} \]

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 2.4e-22)
   (/ (- (* x y) (* z t)) a)
   (fma (- t) (/ z a) (/ (* 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 (a <= 2.4e-22) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = fma(-t, (z / a), ((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 (a <= 2.4e-22)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = fma(Float64(-t), Float64(z / a), Float64(Float64(x * y) / a));
	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_] := If[LessEqual[a, 2.4e-22], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-t) * N[(z / a), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.40000000000000002e-22

   1. Initial program 92.8%

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

   if 2.40000000000000002e-22 < a

   1. Initial program 90.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 90.1

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

   4. Applied rewrites 90.1%

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

Alternative 7: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -20000000000:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

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) -20000000000.0)
   (* (/ 1.0 a) (* x y))
   (if (<= (* x y) 5e+19) (/ (* z (- t)) a) (/ (* 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) <= -20000000000.0) {
		tmp = (1.0 / a) * (x * y);
	} else if ((x * y) <= 5e+19) {
		tmp = (z * -t) / a;
	} 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) <= (-20000000000.0d0)) then
        tmp = (1.0d0 / a) * (x * y)
    else if ((x * y) <= 5d+19) then
        tmp = (z * -t) / a
    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) <= -20000000000.0) {
		tmp = (1.0 / a) * (x * y);
	} else if ((x * y) <= 5e+19) {
		tmp = (z * -t) / a;
	} 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) <= -20000000000.0:
		tmp = (1.0 / a) * (x * y)
	elif (x * y) <= 5e+19:
		tmp = (z * -t) / a
	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) <= -20000000000.0)
		tmp = Float64(Float64(1.0 / a) * Float64(x * y));
	elseif (Float64(x * y) <= 5e+19)
		tmp = Float64(Float64(z * Float64(-t)) / a);
	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) <= -20000000000.0)
		tmp = (1.0 / a) * (x * y);
	elseif ((x * y) <= 5e+19)
		tmp = (z * -t) / a;
	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], -20000000000.0], N[(N[(1.0 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+19], N[(N[(z * (-t)), $MachinePrecision] / a), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e10

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x \cdot y}}} \]

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2e10 < (*.f64 x y) < 5e19

   1. Initial program 93.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if 5e19 < (*.f64 x y)

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -20000000000:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;x \cdot y \leq -21500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) a)))
   (if (<= (* x y) -21500000000.0)
     t_1
     (if (<= (* x y) 3.5e+19) (/ (* z (- t)) a) t_1))))

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) / a;
	double tmp;
	if ((x * y) <= -21500000000.0) {
		tmp = t_1;
	} else if ((x * y) <= 3.5e+19) {
		tmp = (z * -t) / a;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * y) / a
    if ((x * y) <= (-21500000000.0d0)) then
        tmp = t_1
    else if ((x * y) <= 3.5d+19) then
        tmp = (z * -t) / a
    else
        tmp = t_1
    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 t_1 = (x * y) / a;
	double tmp;
	if ((x * y) <= -21500000000.0) {
		tmp = t_1;
	} else if ((x * y) <= 3.5e+19) {
		tmp = (z * -t) / a;
	} else {
		tmp = t_1;
	}
	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) / a
	tmp = 0
	if (x * y) <= -21500000000.0:
		tmp = t_1
	elif (x * y) <= 3.5e+19:
		tmp = (z * -t) / a
	else:
		tmp = t_1
	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) / a)
	tmp = 0.0
	if (Float64(x * y) <= -21500000000.0)
		tmp = t_1;
	elseif (Float64(x * y) <= 3.5e+19)
		tmp = Float64(Float64(z * Float64(-t)) / a);
	else
		tmp = t_1;
	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) / a;
	tmp = 0.0;
	if ((x * y) <= -21500000000.0)
		tmp = t_1;
	elseif ((x * y) <= 3.5e+19)
		tmp = (z * -t) / a;
	else
		tmp = t_1;
	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] / a), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -21500000000.0], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 3.5e+19], N[(N[(z * (-t)), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.15e10 or 3.5e19 < (*.f64 x y)

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -2.15e10 < (*.f64 x y) < 3.5e19

   1. Initial program 93.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -21500000000:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 1.0× speedup

Math

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

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

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(fma(y, x, Float64(z * Float64(-t))) / a)
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[(y * x + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 92.1

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

4. Applied rewrites 92.1%

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

5. Final simplification 92.1%

   \[\leadsto \frac{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)}{a} \]
6. Add Preprocessing

Alternative 10: 91.6% accurate, 1.0× speedup

Math

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

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

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 * y) - (z * t)) / a
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 * y) - (z * t)) / a;
}

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

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(Float64(x * y) - Float64(z * t)) / a)
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 * y) - (z * t)) / a;
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[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

Alternative 11: 50.8% accurate, 1.5× speedup

Math

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

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 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) {
	return (x * y) / a;
}

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 * y) / a
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 * y) / a;
}

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 * y) / a

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 * y) / a)
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 * y) / a;
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 * y), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 48.8

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

5. Applied rewrites 48.8%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a} \]
6. Add Preprocessing

Developer Target 1: 91.9% 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 2024221 
(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))