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

Percentage Accurate: 91.3% → 94.8%

Time: 8.0s

Alternatives: 7

Speedup: 0.6×

• 27.9% 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: 91.3% 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: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 3.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, \left(-t\right) \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 3.9e+42)
    (/ (- (* x y) (* z t)) a_m)
    (fma (/ y a_m) x (* (- t) (/ z a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 3.9e+42) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma((y / a_m), x, (-t * (z / a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 3.9e+42)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(y / a_m), x, Float64(Float64(-t) * Float64(z / a_m)));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 3.9e+42], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x + N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 3.8999999999999997e42

   1. Initial program 90.9%

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

   if 3.8999999999999997e42 < a

   1. Initial program 73.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 91.9

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

   4. Applied rewrites 91.9%

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

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+268}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{x}{t}, -z\right)}{a\_m} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+268)))
      (* (/ (fma y (/ x t) (- z)) a_m) t)
      (/ t_1 a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+268)) {
		tmp = (fma(y, (x / t), -z) / a_m) * t;
	} else {
		tmp = t_1 / a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+268))
		tmp = Float64(Float64(fma(y, Float64(x / t), Float64(-z)) / a_m) * t);
	else
		tmp = Float64(t_1 / a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+268]], $MachinePrecision]], N[(N[(N[(y * N[(x / t), $MachinePrecision] + (-z)), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], N[(t$95$1 / a$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1.9999999999999999e268 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 57.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 76.6

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 83.6%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e268

   1. Initial program 99.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+268}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{x}{t}, -z\right)}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+304}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+223}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_1 -1e+304)
      (* (/ x a_m) y)
      (if (<= t_1 1e+223) (/ (* x y) a_m) (* (/ y a_m) x))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -1e+304) {
		tmp = (x / a_m) * y;
	} else if (t_1 <= 1e+223) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (y / a_m) * x;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if (t_1 <= (-1d+304)) then
        tmp = (x / a_m) * y
    else if (t_1 <= 1d+223) then
        tmp = (x * y) / a_m
    else
        tmp = (y / a_m) * x
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -1e+304) {
		tmp = (x / a_m) * y;
	} else if (t_1 <= 1e+223) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (y / a_m) * x;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -1e+304:
		tmp = (x / a_m) * y
	elif t_1 <= 1e+223:
		tmp = (x * y) / a_m
	else:
		tmp = (y / a_m) * x
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e+304)
		tmp = Float64(Float64(x / a_m) * y);
	elseif (t_1 <= 1e+223)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = Float64(Float64(y / a_m) * x);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -1e+304)
		tmp = (x / a_m) * y;
	elseif (t_1 <= 1e+223)
		tmp = (x * y) / a_m;
	else
		tmp = (y / a_m) * x;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -1e+304], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+223], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999994e303

   1. Initial program 66.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 55.8

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

   5. Applied rewrites 55.8%

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

   if -9.9999999999999994e303 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000005e223

   1. Initial program 99.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 53.6

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

   8. Applied rewrites 53.6%

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

   if 1.00000000000000005e223 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 61.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 63.1%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+304}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+223}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+266} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= (* x y) -1e+266) (not (<= (* x y) 2e+273)))
    (* (/ y a_m) x)
    (/ (- (* x y) (* z t)) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -1e+266) || !((x * y) <= 2e+273)) {
		tmp = (y / a_m) * x;
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (((x * y) <= (-1d+266)) .or. (.not. ((x * y) <= 2d+273))) then
        tmp = (y / a_m) * x
    else
        tmp = ((x * y) - (z * t)) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((x * y) <= -1e+266) || !((x * y) <= 2e+273)) {
		tmp = (y / a_m) * x;
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if ((x * y) <= -1e+266) or not ((x * y) <= 2e+273):
		tmp = (y / a_m) * x
	else:
		tmp = ((x * y) - (z * t)) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+266) || !(Float64(x * y) <= 2e+273))
		tmp = Float64(Float64(y / a_m) * x);
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) <= -1e+266) || ~(((x * y) <= 2e+273)))
		tmp = (y / a_m) * x;
	else
		tmp = ((x * y) - (z * t)) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+266], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+273]], $MachinePrecision]], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e266 or 1.99999999999999989e273 < (*.f64 x y)

   1. Initial program 61.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 95.8%

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

   if -1e266 < (*.f64 x y) < 1.99999999999999989e273

   1. Initial program 93.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+266} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -1e-37)
    (* (/ y a_m) x)
    (if (<= (* x y) 2e-6) (* (- t) (/ z a_m)) (* (/ x a_m) y)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-37) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 2e-6) {
		tmp = -t * (z / a_m);
	} else {
		tmp = (x / a_m) * y;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-1d-37)) then
        tmp = (y / a_m) * x
    else if ((x * y) <= 2d-6) then
        tmp = -t * (z / a_m)
    else
        tmp = (x / a_m) * y
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e-37) {
		tmp = (y / a_m) * x;
	} else if ((x * y) <= 2e-6) {
		tmp = -t * (z / a_m);
	} else {
		tmp = (x / a_m) * y;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -1e-37:
		tmp = (y / a_m) * x
	elif (x * y) <= 2e-6:
		tmp = -t * (z / a_m)
	else:
		tmp = (x / a_m) * y
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -1e-37)
		tmp = Float64(Float64(y / a_m) * x);
	elseif (Float64(x * y) <= 2e-6)
		tmp = Float64(Float64(-t) * Float64(z / a_m));
	else
		tmp = Float64(Float64(x / a_m) * y);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -1e-37)
		tmp = (y / a_m) * x;
	elseif ((x * y) <= 2e-6)
		tmp = -t * (z / a_m);
	else
		tmp = (x / a_m) * y;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -1e-37], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-6], N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000007e-37

   1. Initial program 86.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 75.0%

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

   if -1.00000000000000007e-37 < (*.f64 x y) < 1.99999999999999991e-6

   1. Initial program 92.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if 1.99999999999999991e-6 < (*.f64 x y)

   1. Initial program 76.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (* a_s (if (<= z -4e-112) (* (/ x a_m) y) (* (/ y a_m) x))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (z <= -4e-112) {
		tmp = (x / a_m) * y;
	} else {
		tmp = (y / a_m) * x;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (z <= (-4d-112)) then
        tmp = (x / a_m) * y
    else
        tmp = (y / a_m) * x
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (z <= -4e-112) {
		tmp = (x / a_m) * y;
	} else {
		tmp = (y / a_m) * x;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if z <= -4e-112:
		tmp = (x / a_m) * y
	else:
		tmp = (y / a_m) * x
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (z <= -4e-112)
		tmp = Float64(Float64(x / a_m) * y);
	else
		tmp = Float64(Float64(y / a_m) * x);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (z <= -4e-112)
		tmp = (x / a_m) * y;
	else
		tmp = (y / a_m) * x;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[z, -4e-112], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.9999999999999998e-112

   1. Initial program 89.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 46.6

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

   5. Applied rewrites 46.6%

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

   if -3.9999999999999998e-112 < z

   1. Initial program 85.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 54.1

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

   5. Applied rewrites 54.1%

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

   7. Applied rewrites 56.6%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(\frac{x}{a\_m} \cdot y\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (* (/ x a_m) y)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x / a_m) * y);
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    code = a_s * ((x / a_m) * y)
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x / a_m) * y);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * ((x / a_m) * y)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(x / a_m) * y))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * ((x / a_m) * y);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.0%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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 51.6

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

5. Applied rewrites 51.6%

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

6. Final simplification 51.6%

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

Developer Target 1: 90.9% 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 2024326 
(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))