Quotient of products

Percentage Accurate: 86.6% → 99.1%

Time: 1.9s

Alternatives: 7

Speedup: 1.0×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a2}{b1 \cdot b2} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 * a2) / (b1 * b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 * a2) / Float64(b1 * b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 * a2) / (b1 * b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]

Initial Program: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a2}{b1 \cdot b2} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 * a2) / (b1 * b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 * a2) / Float64(b1 * b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 * a2) / (b1 * b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 5 \cdot 10^{-312}:\\ \;\;\;\;\frac{\frac{a1\_m}{b2\_m}}{b1\_m} \cdot a2\_m\\ \mathbf{elif}\;b1\_m \cdot b2\_m \leq 50000000000000:\\ \;\;\;\;\frac{a1\_m}{b1\_m \cdot b2\_m} \cdot a2\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2\_m}{b2\_m}}{b1\_m} \cdot a1\_m\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (*
  a1_s
  (*
   a2_s
   (*
    b1_s
    (*
     b2_s
     (if (<= (* b1_m b2_m) 5e-312)
       (* (/ (/ a1_m b2_m) b1_m) a2_m)
       (if (<= (* b1_m b2_m) 50000000000000.0)
         (* (/ a1_m (* b1_m b2_m)) a2_m)
         (* (/ (/ a2_m b2_m) b1_m) a1_m))))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((b1_m * b2_m) <= 5e-312) {
		tmp = ((a1_m / b2_m) / b1_m) * a2_m;
	} else if ((b1_m * b2_m) <= 50000000000000.0) {
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	} else {
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    real(8) :: tmp
    if ((b1_m * b2_m) <= 5d-312) then
        tmp = ((a1_m / b2_m) / b1_m) * a2_m
    else if ((b1_m * b2_m) <= 50000000000000.0d0) then
        tmp = (a1_m / (b1_m * b2_m)) * a2_m
    else
        tmp = ((a2_m / b2_m) / b1_m) * a1_m
    end if
    code = a1_s * (a2_s * (b1_s * (b2_s * tmp)))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((b1_m * b2_m) <= 5e-312) {
		tmp = ((a1_m / b2_m) / b1_m) * a2_m;
	} else if ((b1_m * b2_m) <= 50000000000000.0) {
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	} else {
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	tmp = 0
	if (b1_m * b2_m) <= 5e-312:
		tmp = ((a1_m / b2_m) / b1_m) * a2_m
	elif (b1_m * b2_m) <= 50000000000000.0:
		tmp = (a1_m / (b1_m * b2_m)) * a2_m
	else:
		tmp = ((a2_m / b2_m) / b1_m) * a1_m
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0
	if (Float64(b1_m * b2_m) <= 5e-312)
		tmp = Float64(Float64(Float64(a1_m / b2_m) / b1_m) * a2_m);
	elseif (Float64(b1_m * b2_m) <= 50000000000000.0)
		tmp = Float64(Float64(a1_m / Float64(b1_m * b2_m)) * a2_m);
	else
		tmp = Float64(Float64(Float64(a2_m / b2_m) / b1_m) * a1_m);
	end
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * tmp))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp_2 = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0;
	if ((b1_m * b2_m) <= 5e-312)
		tmp = ((a1_m / b2_m) / b1_m) * a2_m;
	elseif ((b1_m * b2_m) <= 50000000000000.0)
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	else
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	end
	tmp_2 = a1_s * (a2_s * (b1_s * (b2_s * tmp)));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 5e-312], N[(N[(N[(a1$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision] * a2$95$m), $MachinePrecision], If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 50000000000000.0], N[(N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision], N[(N[(N[(a2$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision] * a1$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < 5.0000000000022e-312

   1. Initial program 86.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      6. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      9. lower-/.f64 89.5

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

   4. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1} \cdot a2} \]

   if 5.0000000000022e-312 < (*.f64 b1 b2) < 5e13

   1. Initial program 94.9%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      6. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      9. lower-/.f64 86.9

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1} \cdot a2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

      3. remove-double-neg N/A

         \[\leadsto \frac{\frac{a1}{b2}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)}} \cdot a2 \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{a1}{b2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right)}} \cdot a2 \]

      5. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      8. remove-double-neg 93.4

         \[\leadsto \frac{a1}{\color{blue}{b1} \cdot b2} \cdot a2 \]

   6. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

   if 5e13 < (*.f64 b1 b2)

   1. Initial program 86.1%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      7. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      10. lower-/.f64 84.7

          \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

   4. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ \begin{array}{l} t_0 := \frac{a1\_m \cdot a2\_m}{b1\_m \cdot b2\_m}\\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a2\_m}{b1\_m} \cdot \frac{a1\_m}{b2\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1\_m}{b2\_m}}{b1\_m} \cdot a2\_m\\ \end{array}\right)\right)\right) \end{array} \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (let* ((t_0 (/ (* a1_m a2_m) (* b1_m b2_m))))
   (*
    a1_s
    (*
     a2_s
     (*
      b1_s
      (*
       b2_s
       (if (<= t_0 0.0)
         (* (/ a2_m b1_m) (/ a1_m b2_m))
         (if (<= t_0 2e+232) t_0 (* (/ (/ a1_m b2_m) b1_m) a2_m)))))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double t_0 = (a1_m * a2_m) / (b1_m * b2_m);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (a2_m / b1_m) * (a1_m / b2_m);
	} else if (t_0 <= 2e+232) {
		tmp = t_0;
	} else {
		tmp = ((a1_m / b2_m) / b1_m) * a2_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a1_m * a2_m) / (b1_m * b2_m)
    if (t_0 <= 0.0d0) then
        tmp = (a2_m / b1_m) * (a1_m / b2_m)
    else if (t_0 <= 2d+232) then
        tmp = t_0
    else
        tmp = ((a1_m / b2_m) / b1_m) * a2_m
    end if
    code = a1_s * (a2_s * (b1_s * (b2_s * tmp)))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double t_0 = (a1_m * a2_m) / (b1_m * b2_m);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (a2_m / b1_m) * (a1_m / b2_m);
	} else if (t_0 <= 2e+232) {
		tmp = t_0;
	} else {
		tmp = ((a1_m / b2_m) / b1_m) * a2_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	t_0 = (a1_m * a2_m) / (b1_m * b2_m)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (a2_m / b1_m) * (a1_m / b2_m)
	elif t_0 <= 2e+232:
		tmp = t_0
	else:
		tmp = ((a1_m / b2_m) / b1_m) * a2_m
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	t_0 = Float64(Float64(a1_m * a2_m) / Float64(b1_m * b2_m))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(a2_m / b1_m) * Float64(a1_m / b2_m));
	elseif (t_0 <= 2e+232)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a1_m / b2_m) / b1_m) * a2_m);
	end
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * tmp))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp_2 = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	t_0 = (a1_m * a2_m) / (b1_m * b2_m);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (a2_m / b1_m) * (a1_m / b2_m);
	elseif (t_0 <= 2e+232)
		tmp = t_0;
	else
		tmp = ((a1_m / b2_m) / b1_m) * a2_m;
	end
	tmp_2 = a1_s * (a2_s * (b1_s * (b2_s * tmp)));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := Block[{t$95$0 = N[(N[(a1$95$m * a2$95$m), $MachinePrecision] / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(a2$95$m / b1$95$m), $MachinePrecision] * N[(a1$95$m / b2$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+232], t$95$0, N[(N[(N[(a1$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision] * a2$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 89.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a1}}{b1 \cdot b2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{a2 \cdot a1}{\color{blue}{b1 \cdot b2}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1}} \cdot \frac{a1}{b2} \]

      8. lower-/.f64 91.0

         \[\leadsto \frac{a2}{b1} \cdot \color{blue}{\frac{a1}{b2}} \]

   4. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

   if 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.00000000000000011e232

   1. Initial program 97.4%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing

   if 2.00000000000000011e232 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 72.1%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      6. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      9. lower-/.f64 88.5

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

   4. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1} \cdot a2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ \begin{array}{l} t_0 := \frac{a2\_m}{b1\_m} \cdot \frac{a1\_m}{b2\_m}\\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 5 \cdot 10^{-312}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b1\_m \cdot b2\_m \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\frac{a1\_m}{b1\_m \cdot b2\_m} \cdot a2\_m\\ \mathbf{elif}\;b1\_m \cdot b2\_m \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{a2\_m}{b1\_m \cdot b2\_m} \cdot a1\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right)\right)\right) \end{array} \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (let* ((t_0 (* (/ a2_m b1_m) (/ a1_m b2_m))))
   (*
    a1_s
    (*
     a2_s
     (*
      b1_s
      (*
       b2_s
       (if (<= (* b1_m b2_m) 5e-312)
         t_0
         (if (<= (* b1_m b2_m) 2e-31)
           (* (/ a1_m (* b1_m b2_m)) a2_m)
           (if (<= (* b1_m b2_m) 2e+301)
             (* (/ a2_m (* b1_m b2_m)) a1_m)
             t_0)))))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double t_0 = (a2_m / b1_m) * (a1_m / b2_m);
	double tmp;
	if ((b1_m * b2_m) <= 5e-312) {
		tmp = t_0;
	} else if ((b1_m * b2_m) <= 2e-31) {
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	} else if ((b1_m * b2_m) <= 2e+301) {
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	} else {
		tmp = t_0;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a2_m / b1_m) * (a1_m / b2_m)
    if ((b1_m * b2_m) <= 5d-312) then
        tmp = t_0
    else if ((b1_m * b2_m) <= 2d-31) then
        tmp = (a1_m / (b1_m * b2_m)) * a2_m
    else if ((b1_m * b2_m) <= 2d+301) then
        tmp = (a2_m / (b1_m * b2_m)) * a1_m
    else
        tmp = t_0
    end if
    code = a1_s * (a2_s * (b1_s * (b2_s * tmp)))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double t_0 = (a2_m / b1_m) * (a1_m / b2_m);
	double tmp;
	if ((b1_m * b2_m) <= 5e-312) {
		tmp = t_0;
	} else if ((b1_m * b2_m) <= 2e-31) {
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	} else if ((b1_m * b2_m) <= 2e+301) {
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	} else {
		tmp = t_0;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	t_0 = (a2_m / b1_m) * (a1_m / b2_m)
	tmp = 0
	if (b1_m * b2_m) <= 5e-312:
		tmp = t_0
	elif (b1_m * b2_m) <= 2e-31:
		tmp = (a1_m / (b1_m * b2_m)) * a2_m
	elif (b1_m * b2_m) <= 2e+301:
		tmp = (a2_m / (b1_m * b2_m)) * a1_m
	else:
		tmp = t_0
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	t_0 = Float64(Float64(a2_m / b1_m) * Float64(a1_m / b2_m))
	tmp = 0.0
	if (Float64(b1_m * b2_m) <= 5e-312)
		tmp = t_0;
	elseif (Float64(b1_m * b2_m) <= 2e-31)
		tmp = Float64(Float64(a1_m / Float64(b1_m * b2_m)) * a2_m);
	elseif (Float64(b1_m * b2_m) <= 2e+301)
		tmp = Float64(Float64(a2_m / Float64(b1_m * b2_m)) * a1_m);
	else
		tmp = t_0;
	end
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * tmp))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp_2 = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	t_0 = (a2_m / b1_m) * (a1_m / b2_m);
	tmp = 0.0;
	if ((b1_m * b2_m) <= 5e-312)
		tmp = t_0;
	elseif ((b1_m * b2_m) <= 2e-31)
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	elseif ((b1_m * b2_m) <= 2e+301)
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	else
		tmp = t_0;
	end
	tmp_2 = a1_s * (a2_s * (b1_s * (b2_s * tmp)));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := Block[{t$95$0 = N[(N[(a2$95$m / b1$95$m), $MachinePrecision] * N[(a1$95$m / b2$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 5e-312], t$95$0, If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 2e-31], N[(N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision], If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 2e+301], N[(N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < 5.0000000000022e-312 or 2.00000000000000011e301 < (*.f64 b1 b2)

   1. Initial program 85.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{a2 \cdot a1}}{b1 \cdot b2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{a2 \cdot a1}{\color{blue}{b1 \cdot b2}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1}} \cdot \frac{a1}{b2} \]

      8. lower-/.f64 91.1

         \[\leadsto \frac{a2}{b1} \cdot \color{blue}{\frac{a1}{b2}} \]

   4. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

   if 5.0000000000022e-312 < (*.f64 b1 b2) < 2e-31

   1. Initial program 94.1%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      6. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      9. lower-/.f64 87.3

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

   4. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1} \cdot a2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

      3. remove-double-neg N/A

         \[\leadsto \frac{\frac{a1}{b2}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)}} \cdot a2 \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{a1}{b2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right)}} \cdot a2 \]

      5. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      8. remove-double-neg 92.3

         \[\leadsto \frac{a1}{\color{blue}{b1} \cdot b2} \cdot a2 \]

   6. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

   if 2e-31 < (*.f64 b1 b2) < 2.00000000000000011e301

   1. Initial program 91.5%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      7. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      10. lower-/.f64 81.1

          \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

   4. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

      3. remove-double-neg N/A

         \[\leadsto \frac{\frac{a2}{b2}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)}} \cdot a1 \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{a2}{b2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right)}} \cdot a1 \]

      5. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a1 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a1 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a1 \]

      8. remove-double-neg 93.7

         \[\leadsto \frac{a2}{\color{blue}{b1} \cdot b2} \cdot a1 \]

   6. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2}} \cdot a1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 50000000000000:\\ \;\;\;\;\frac{\frac{a1\_m}{b1\_m}}{b2\_m} \cdot a2\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2\_m}{b2\_m}}{b1\_m} \cdot a1\_m\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (*
  a1_s
  (*
   a2_s
   (*
    b1_s
    (*
     b2_s
     (if (<= (* b1_m b2_m) 50000000000000.0)
       (* (/ (/ a1_m b1_m) b2_m) a2_m)
       (* (/ (/ a2_m b2_m) b1_m) a1_m)))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((b1_m * b2_m) <= 50000000000000.0) {
		tmp = ((a1_m / b1_m) / b2_m) * a2_m;
	} else {
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    real(8) :: tmp
    if ((b1_m * b2_m) <= 50000000000000.0d0) then
        tmp = ((a1_m / b1_m) / b2_m) * a2_m
    else
        tmp = ((a2_m / b2_m) / b1_m) * a1_m
    end if
    code = a1_s * (a2_s * (b1_s * (b2_s * tmp)))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((b1_m * b2_m) <= 50000000000000.0) {
		tmp = ((a1_m / b1_m) / b2_m) * a2_m;
	} else {
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	tmp = 0
	if (b1_m * b2_m) <= 50000000000000.0:
		tmp = ((a1_m / b1_m) / b2_m) * a2_m
	else:
		tmp = ((a2_m / b2_m) / b1_m) * a1_m
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0
	if (Float64(b1_m * b2_m) <= 50000000000000.0)
		tmp = Float64(Float64(Float64(a1_m / b1_m) / b2_m) * a2_m);
	else
		tmp = Float64(Float64(Float64(a2_m / b2_m) / b1_m) * a1_m);
	end
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * tmp))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp_2 = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0;
	if ((b1_m * b2_m) <= 50000000000000.0)
		tmp = ((a1_m / b1_m) / b2_m) * a2_m;
	else
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	end
	tmp_2 = a1_s * (a2_s * (b1_s * (b2_s * tmp)));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 50000000000000.0], N[(N[(N[(a1$95$m / b1$95$m), $MachinePrecision] / b2$95$m), $MachinePrecision] * a2$95$m), $MachinePrecision], N[(N[(N[(a2$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision] * a1$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b1 b2) < 5e13

   1. Initial program 88.2%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(a1 \cdot a2\right)}{\mathsf{neg}\left(b1 \cdot b2\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{a1 \cdot a2}\right)}{\mathsf{neg}\left(b1 \cdot b2\right)} \]

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

         \[\leadsto \frac{\color{blue}{a1 \cdot \left(\mathsf{neg}\left(a2\right)\right)}}{\mathsf{neg}\left(b1 \cdot b2\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{a1}{\mathsf{neg}\left(b1 \cdot b2\right)} \cdot \left(\mathsf{neg}\left(a2\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{\mathsf{neg}\left(b1 \cdot b2\right)} \cdot \left(\mathsf{neg}\left(a2\right)\right)} \]

      7. frac-2neg-rev N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(a1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1 \cdot b2\right)\right)\right)}} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a1\right)}{\color{blue}{b1 \cdot b2}} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(a1\right)}{\color{blue}{b1 \cdot b2}} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(a1\right)}{b1}}{b2}} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(a1\right)}{b1}}{b2}} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(a1\right)}{b1}}}{b2} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-a1}}{b1}}{b2} \cdot \left(\mathsf{neg}\left(a2\right)\right) \]

      14. lower-neg.f64 87.1

          \[\leadsto \frac{\frac{-a1}{b1}}{b2} \cdot \color{blue}{\left(-a2\right)} \]

   4. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{\frac{-a1}{b1}}{b2} \cdot \left(-a2\right)} \]

   if 5e13 < (*.f64 b1 b2)

   1. Initial program 86.1%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      7. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      10. lower-/.f64 84.7

          \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

   4. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq 50000000000000:\\ \;\;\;\;\frac{\frac{a1}{b1}}{b2} \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;a1\_m \cdot a2\_m \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{a1\_m}{b1\_m} \cdot a2\_m}{b2\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2\_m}{b2\_m}}{b1\_m} \cdot a1\_m\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (*
  a1_s
  (*
   a2_s
   (*
    b1_s
    (*
     b2_s
     (if (<= (* a1_m a2_m) 2e+87)
       (/ (* (/ a1_m b1_m) a2_m) b2_m)
       (* (/ (/ a2_m b2_m) b1_m) a1_m)))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((a1_m * a2_m) <= 2e+87) {
		tmp = ((a1_m / b1_m) * a2_m) / b2_m;
	} else {
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    real(8) :: tmp
    if ((a1_m * a2_m) <= 2d+87) then
        tmp = ((a1_m / b1_m) * a2_m) / b2_m
    else
        tmp = ((a2_m / b2_m) / b1_m) * a1_m
    end if
    code = a1_s * (a2_s * (b1_s * (b2_s * tmp)))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((a1_m * a2_m) <= 2e+87) {
		tmp = ((a1_m / b1_m) * a2_m) / b2_m;
	} else {
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	tmp = 0
	if (a1_m * a2_m) <= 2e+87:
		tmp = ((a1_m / b1_m) * a2_m) / b2_m
	else:
		tmp = ((a2_m / b2_m) / b1_m) * a1_m
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0
	if (Float64(a1_m * a2_m) <= 2e+87)
		tmp = Float64(Float64(Float64(a1_m / b1_m) * a2_m) / b2_m);
	else
		tmp = Float64(Float64(Float64(a2_m / b2_m) / b1_m) * a1_m);
	end
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * tmp))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp_2 = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0;
	if ((a1_m * a2_m) <= 2e+87)
		tmp = ((a1_m / b1_m) * a2_m) / b2_m;
	else
		tmp = ((a2_m / b2_m) / b1_m) * a1_m;
	end
	tmp_2 = a1_s * (a2_s * (b1_s * (b2_s * tmp)));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * If[LessEqual[N[(a1$95$m * a2$95$m), $MachinePrecision], 2e+87], N[(N[(N[(a1$95$m / b1$95$m), $MachinePrecision] * a2$95$m), $MachinePrecision] / b2$95$m), $MachinePrecision], N[(N[(N[(a2$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision] * a1$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a2) < 1.9999999999999999e87

   1. Initial program 88.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{a1 \cdot a2}{\color{blue}{b1 \cdot b2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{a1 \cdot a2}}{b1}}{b2} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{a1}{b1} \cdot a2}}{b2} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a1}{b1} \cdot a2}}{b2} \]

      8. lower-/.f64 85.1

         \[\leadsto \frac{\color{blue}{\frac{a1}{b1}} \cdot a2}{b2} \]

   4. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1} \cdot a2}{b2}} \]

   if 1.9999999999999999e87 < (*.f64 a1 a2)

   1. Initial program 84.7%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      7. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      10. lower-/.f64 87.9

          \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 10^{-29}:\\ \;\;\;\;\frac{a1\_m}{b1\_m \cdot b2\_m} \cdot a2\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{a2\_m}{b1\_m \cdot b2\_m} \cdot a1\_m\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (*
  a1_s
  (*
   a2_s
   (*
    b1_s
    (*
     b2_s
     (if (<= (* b1_m b2_m) 1e-29)
       (* (/ a1_m (* b1_m b2_m)) a2_m)
       (* (/ a2_m (* b1_m b2_m)) a1_m)))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((b1_m * b2_m) <= 1e-29) {
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	} else {
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    real(8) :: tmp
    if ((b1_m * b2_m) <= 1d-29) then
        tmp = (a1_m / (b1_m * b2_m)) * a2_m
    else
        tmp = (a2_m / (b1_m * b2_m)) * a1_m
    end if
    code = a1_s * (a2_s * (b1_s * (b2_s * tmp)))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if ((b1_m * b2_m) <= 1e-29) {
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	} else {
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	}
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	tmp = 0
	if (b1_m * b2_m) <= 1e-29:
		tmp = (a1_m / (b1_m * b2_m)) * a2_m
	else:
		tmp = (a2_m / (b1_m * b2_m)) * a1_m
	return a1_s * (a2_s * (b1_s * (b2_s * tmp)))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0
	if (Float64(b1_m * b2_m) <= 1e-29)
		tmp = Float64(Float64(a1_m / Float64(b1_m * b2_m)) * a2_m);
	else
		tmp = Float64(Float64(a2_m / Float64(b1_m * b2_m)) * a1_m);
	end
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * tmp))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp_2 = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0;
	if ((b1_m * b2_m) <= 1e-29)
		tmp = (a1_m / (b1_m * b2_m)) * a2_m;
	else
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	end
	tmp_2 = a1_s * (a2_s * (b1_s * (b2_s * tmp)));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 1e-29], N[(N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision], N[(N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b1 b2) < 9.99999999999999943e-30

   1. Initial program 87.9%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

      6. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      9. lower-/.f64 89.0

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

   4. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1} \cdot a2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

      3. remove-double-neg N/A

         \[\leadsto \frac{\frac{a1}{b2}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)}} \cdot a2 \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{a1}{b2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right)}} \cdot a2 \]

      5. *-commutative N/A

         \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

      8. remove-double-neg 87.6

         \[\leadsto \frac{a1}{\color{blue}{b1} \cdot b2} \cdot a2 \]

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]

   if 9.99999999999999943e-30 < (*.f64 b1 b2)

   1. Initial program 87.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2} \cdot a1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

      7. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{b2 \cdot b1}} \cdot a1 \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      10. lower-/.f64 84.7

          \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

   4. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1}} \cdot a1 \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{b1} \cdot a1 \]

      3. remove-double-neg N/A

         \[\leadsto \frac{\frac{a2}{b2}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)}} \cdot a1 \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{a2}{b2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right)}} \cdot a1 \]

      5. *-commutative N/A

         \[\leadsto \frac{a2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a1 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a2}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a1 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{a2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a1 \]

      8. remove-double-neg 88.9

         \[\leadsto \frac{a2}{\color{blue}{b1} \cdot b2} \cdot a1 \]

   6. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{a2}{b1 \cdot b2}} \cdot a1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \left(\frac{a1\_m}{b1\_m \cdot b2\_m} \cdot a2\_m\right)\right)\right)\right) \end{array} \]

FPCore

b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
(FPCore (a1_s a2_s b1_s b2_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (* a1_s (* a2_s (* b1_s (* b2_s (* (/ a1_m (* b1_m b2_m)) a2_m))))))

C

b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
assert(a1_m < a2_m && a2_m < b1_m && b1_m < b2_m);
double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	return a1_s * (a2_s * (b1_s * (b2_s * ((a1_m / (b1_m * b2_m)) * a2_m))));
}

Fortran

b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
real(8) function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: a1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: b1_m
    real(8), intent (in) :: b2_m
    code = a1_s * (a2_s * (b1_s * (b2_s * ((a1_m / (b1_m * b2_m)) * a2_m))))
end function

Java

b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
assert a1_m < a2_m && a2_m < b1_m && b1_m < b2_m;
public static double code(double a1_s, double a2_s, double b1_s, double b2_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	return a1_s * (a2_s * (b1_s * (b2_s * ((a1_m / (b1_m * b2_m)) * a2_m))));
}

Python

b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
[a1_m, a2_m, b1_m, b2_m] = sort([a1_m, a2_m, b1_m, b2_m])
def code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m):
	return a1_s * (a2_s * (b1_s * (b2_s * ((a1_m / (b1_m * b2_m)) * a2_m))))

Julia

b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
a1_m, a2_m, b1_m, b2_m = sort([a1_m, a2_m, b1_m, b2_m])
function code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	return Float64(a1_s * Float64(a2_s * Float64(b1_s * Float64(b2_s * Float64(Float64(a1_m / Float64(b1_m * b2_m)) * a2_m)))))
end

MATLAB

b2\_m = abs(b2);
b2\_s = sign(b2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
a1_m, a2_m, b1_m, b2_m = num2cell(sort([a1_m, a2_m, b1_m, b2_m])){:}
function tmp = code(a1_s, a2_s, b1_s, b2_s, a1_m, a2_m, b1_m, b2_m)
	tmp = a1_s * (a2_s * (b1_s * (b2_s * ((a1_m / (b1_m * b2_m)) * a2_m))));
end

Wolfram

b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a2\_m = N[Abs[a2], $MachinePrecision]
a2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
NOTE: a1_m, a2_m, b1_m, and b2_m should be sorted in increasing order before calling this function.
code[a1$95$s_, a2$95$s_, b1$95$s_, b2$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(a1$95$s * N[(a2$95$s * N[(b1$95$s * N[(b2$95$s * N[(N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.7%

   \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a2}}{b1 \cdot b2} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2}} \cdot a2 \]

   6. *-commutative N/A

      \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1}} \cdot a2 \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

   9. lower-/.f64 90.1

      \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

4. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1} \cdot a2} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{b1}} \cdot a2 \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{a1}{b2}}}{b1} \cdot a2 \]

   3. remove-double-neg N/A

      \[\leadsto \frac{\frac{a1}{b2}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)}} \cdot a2 \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{a1}{b2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right)}} \cdot a2 \]

   5. *-commutative N/A

      \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{a1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b1\right)\right)\right)\right) \cdot b2}} \cdot a2 \]

   8. remove-double-neg 88.7

      \[\leadsto \frac{a1}{\color{blue}{b1} \cdot b2} \cdot a2 \]

6. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2}} \cdot a2 \]
7. Add Preprocessing

Developer Target 1: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{a1}{b1} \cdot \frac{a2}{b2} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (* (/ a1 b1) (/ a2 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 / b1) * (a2 / b2);
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 / b1) * (a2 / b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 / b1) * (a2 / b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 / b1) * (a2 / b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 / b1) * Float64(a2 / b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 / b1) * (a2 / b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024332 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ a1 b1) (/ a2 b2)))

  (/ (* a1 a2) (* b1 b2)))