Quotient of products

Percentage Accurate: 86.4% → 98.8%

Time: 5.4s

Alternatives: 5

Speedup: 1.0×

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: 86.4% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;a1\_m \cdot a2\_m \leq 10^{-45}:\\ \;\;\;\;\frac{a2\_m \cdot \frac{a1\_m}{b1\_m}}{b2\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1\_m \cdot \frac{a2\_m}{b2\_m}}{b1\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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) 1e-45)
       (/ (* a2_m (/ a1_m b1_m)) b2_m)
       (/ (* a1_m (/ a2_m b2_m)) b1_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) <= 1e-45) {
		tmp = (a2_m * (a1_m / b1_m)) / b2_m;
	} else {
		tmp = (a1_m * (a2_m / b2_m)) / b1_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) <= 1d-45) then
        tmp = (a2_m * (a1_m / b1_m)) / b2_m
    else
        tmp = (a1_m * (a2_m / b2_m)) / b1_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) <= 1e-45) {
		tmp = (a2_m * (a1_m / b1_m)) / b2_m;
	} else {
		tmp = (a1_m * (a2_m / b2_m)) / b1_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) <= 1e-45:
		tmp = (a2_m * (a1_m / b1_m)) / b2_m
	else:
		tmp = (a1_m * (a2_m / b2_m)) / b1_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) <= 1e-45)
		tmp = Float64(Float64(a2_m * Float64(a1_m / b1_m)) / b2_m);
	else
		tmp = Float64(Float64(a1_m * Float64(a2_m / b2_m)) / b1_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) <= 1e-45)
		tmp = (a2_m * (a1_m / b1_m)) / b2_m;
	else
		tmp = (a1_m * (a2_m / b2_m)) / b1_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], 1e-45], N[(N[(a2$95$m * N[(a1$95$m / b1$95$m), $MachinePrecision]), $MachinePrecision] / b2$95$m), $MachinePrecision], N[(N[(a1$95$m * N[(a2$95$m / b2$95$m), $MachinePrecision]), $MachinePrecision] / b1$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a2) < 9.99999999999999984e-46

   1. Initial program 86.0%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   4. Applied rewrites 86.9%

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

   if 9.99999999999999984e-46 < (*.f64 a1 a2)

   1. Initial program 91.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. frac-times N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 91.4

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

   6. Applied rewrites 91.4%

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.4

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

   8. Applied rewrites 84.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq 10^{-45}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ \begin{array}{l} t_0 := \frac{a2\_m}{b1\_m} \cdot \frac{a1\_m}{b2\_m}\\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b1\_m \cdot b2\_m \leq 5 \cdot 10^{+22}:\\ \;\;\;\;a2\_m \cdot \frac{a1\_m}{b1\_m \cdot b2\_m}\\ \mathbf{elif}\;b1\_m \cdot b2\_m \leq 5 \cdot 10^{+298}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b1\_m \cdot b2\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right)\right)\right) \end{array} \end{array} \]

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) 1e-321)
         t_0
         (if (<= (* b1_m b2_m) 5e+22)
           (* a2_m (/ a1_m (* b1_m b2_m)))
           (if (<= (* b1_m b2_m) 5e+298)
             (* a1_m (/ a2_m (* b1_m b2_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) <= 1e-321) {
		tmp = t_0;
	} else if ((b1_m * b2_m) <= 5e+22) {
		tmp = a2_m * (a1_m / (b1_m * b2_m));
	} else if ((b1_m * b2_m) <= 5e+298) {
		tmp = a1_m * (a2_m / (b1_m * b2_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) <= 1d-321) then
        tmp = t_0
    else if ((b1_m * b2_m) <= 5d+22) then
        tmp = a2_m * (a1_m / (b1_m * b2_m))
    else if ((b1_m * b2_m) <= 5d+298) then
        tmp = a1_m * (a2_m / (b1_m * b2_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) <= 1e-321) {
		tmp = t_0;
	} else if ((b1_m * b2_m) <= 5e+22) {
		tmp = a2_m * (a1_m / (b1_m * b2_m));
	} else if ((b1_m * b2_m) <= 5e+298) {
		tmp = a1_m * (a2_m / (b1_m * b2_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) <= 1e-321:
		tmp = t_0
	elif (b1_m * b2_m) <= 5e+22:
		tmp = a2_m * (a1_m / (b1_m * b2_m))
	elif (b1_m * b2_m) <= 5e+298:
		tmp = a1_m * (a2_m / (b1_m * b2_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) <= 1e-321)
		tmp = t_0;
	elseif (Float64(b1_m * b2_m) <= 5e+22)
		tmp = Float64(a2_m * Float64(a1_m / Float64(b1_m * b2_m)));
	elseif (Float64(b1_m * b2_m) <= 5e+298)
		tmp = Float64(a1_m * Float64(a2_m / Float64(b1_m * b2_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) <= 1e-321)
		tmp = t_0;
	elseif ((b1_m * b2_m) <= 5e+22)
		tmp = a2_m * (a1_m / (b1_m * b2_m));
	elseif ((b1_m * b2_m) <= 5e+298)
		tmp = a1_m * (a2_m / (b1_m * b2_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], 1e-321], t$95$0, If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 5e+22], N[(a2$95$m * N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 5e+298], N[(a1$95$m * N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < 9.98013e-322 or 5.0000000000000003e298 < (*.f64 b1 b2)

   1. Initial program 81.8%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

   if 9.98013e-322 < (*.f64 b1 b2) < 4.9999999999999996e22

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 4.9999999999999996e22 < (*.f64 b1 b2) < 5.0000000000000003e298

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq 10^{-321}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{+22}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a1\_m \cdot a2\_m}{b1\_m \cdot b2\_m} \leq 10^{-28}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b1\_m \cdot b2\_m}\\ \mathbf{else}:\\ \;\;\;\;a2\_m \cdot \frac{a1\_m}{b1\_m \cdot b2\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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) (* b1_m b2_m)) 1e-28)
       (* a1_m (/ a2_m (* b1_m b2_m)))
       (* a2_m (/ a1_m (* b1_m b2_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) / (b1_m * b2_m)) <= 1e-28) {
		tmp = a1_m * (a2_m / (b1_m * b2_m));
	} else {
		tmp = a2_m * (a1_m / (b1_m * b2_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) / (b1_m * b2_m)) <= 1d-28) then
        tmp = a1_m * (a2_m / (b1_m * b2_m))
    else
        tmp = a2_m * (a1_m / (b1_m * b2_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) / (b1_m * b2_m)) <= 1e-28) {
		tmp = a1_m * (a2_m / (b1_m * b2_m));
	} else {
		tmp = a2_m * (a1_m / (b1_m * b2_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) / (b1_m * b2_m)) <= 1e-28:
		tmp = a1_m * (a2_m / (b1_m * b2_m))
	else:
		tmp = a2_m * (a1_m / (b1_m * b2_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(Float64(a1_m * a2_m) / Float64(b1_m * b2_m)) <= 1e-28)
		tmp = Float64(a1_m * Float64(a2_m / Float64(b1_m * b2_m)));
	else
		tmp = Float64(a2_m * Float64(a1_m / Float64(b1_m * b2_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) / (b1_m * b2_m)) <= 1e-28)
		tmp = a1_m * (a2_m / (b1_m * b2_m));
	else
		tmp = a2_m * (a1_m / (b1_m * b2_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[(N[(a1$95$m * a2$95$m), $MachinePrecision] / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision], 1e-28], N[(a1$95$m * N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2$95$m * N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 9.99999999999999971e-29

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 91.6

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

   4. Applied rewrites 91.6%

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

   if 9.99999999999999971e-29 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 79.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 77.6

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

   4. Applied rewrites 77.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{-28}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.8× speedup

Math

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

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) (/ a2_m b2_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) * (a2_m / b2_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) * (a2_m / b2_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) * (a2_m / b2_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) * (a2_m / b2_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 / b1_m) * Float64(a2_m / b2_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) * (a2_m / b2_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 / b1$95$m), $MachinePrecision] * N[(a2$95$m / b2$95$m), $MachinePrecision]), $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. times-frac N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 85.4

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

4. Applied rewrites 85.4%

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

5. Final simplification 85.4%

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

Alternative 5: 86.8% accurate, 1.0× speedup

Math

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

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 (* a2_m (/ a1_m (* b1_m b2_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 * (a2_m * (a1_m / (b1_m * b2_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 * (a2_m * (a1_m / (b1_m * b2_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 * (a2_m * (a1_m / (b1_m * b2_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 * (a2_m * (a1_m / (b1_m * b2_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(a2_m * Float64(a1_m / Float64(b1_m * b2_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 * (a2_m * (a1_m / (b1_m * b2_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[(a2$95$m * N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 84.7

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

4. Applied rewrites 84.7%

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

5. Final simplification 84.7%

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

Developer Target 1: 98.0% 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 2024209 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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