Quotient of products

Percentage Accurate: 86.5% → 97.9%

Time: 3.3s

Alternatives: 3

Speedup: 1.0×

• 20.3% 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: 86.5% 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: 97.9% 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 86.8%

   \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
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. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 85.0

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

4. Applied rewrites 85.0%

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

5. Final simplification 85.0%

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

Alternative 2: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ a1\_s \cdot \left(a2\_s \cdot \left(b1\_s \cdot \left(b2\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a1\_m \cdot a2\_m}{b1\_m \cdot b2\_m} \leq 4 \cdot 10^{-97}:\\ \;\;\;\;\frac{a2\_m}{b1\_m \cdot b2\_m} \cdot a1\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{a1\_m}{b1\_m \cdot b2\_m} \cdot a2\_m\\ \end{array}\right)\right)\right) \end{array} \]

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)) 4e-97)
       (* (/ a2_m (* b1_m b2_m)) a1_m)
       (* (/ 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) {
	double tmp;
	if (((a1_m * a2_m) / (b1_m * b2_m)) <= 4e-97) {
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	} else {
		tmp = (a1_m / (b1_m * b2_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) :: tmp
    if (((a1_m * a2_m) / (b1_m * b2_m)) <= 4d-97) then
        tmp = (a2_m / (b1_m * b2_m)) * a1_m
    else
        tmp = (a1_m / (b1_m * b2_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 tmp;
	if (((a1_m * a2_m) / (b1_m * b2_m)) <= 4e-97) {
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	} else {
		tmp = (a1_m / (b1_m * b2_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):
	tmp = 0
	if ((a1_m * a2_m) / (b1_m * b2_m)) <= 4e-97:
		tmp = (a2_m / (b1_m * b2_m)) * a1_m
	else:
		tmp = (a1_m / (b1_m * b2_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)
	tmp = 0.0
	if (Float64(Float64(a1_m * a2_m) / Float64(b1_m * b2_m)) <= 4e-97)
		tmp = Float64(Float64(a2_m / Float64(b1_m * b2_m)) * a1_m);
	else
		tmp = Float64(Float64(a1_m / Float64(b1_m * b2_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)
	tmp = 0.0;
	if (((a1_m * a2_m) / (b1_m * b2_m)) <= 4e-97)
		tmp = (a2_m / (b1_m * b2_m)) * a1_m;
	else
		tmp = (a1_m / (b1_m * b2_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_] := 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], 4e-97], N[(N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a1$95$m), $MachinePrecision], N[(N[(a1$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision] * a2$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.00000000000000014e-97

   1. Initial program 86.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   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. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 87.9

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

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.8

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

   6. Applied rewrites 88.8%

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

   if 4.00000000000000014e-97 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 86.9%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   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

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

      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. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 89.0

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

   4. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.2

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in a1 around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.6

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

   9. Applied rewrites 80.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4 \cdot 10^{-97}:\\ \;\;\;\;\frac{a2}{b1 \cdot b2} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot b2} \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% 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 86.8%

   \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
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. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 88.2

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

4. Applied rewrites 88.2%

   \[\leadsto \color{blue}{\frac{\frac{a2}{b2}}{b1} \cdot a1} \]
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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 87.8

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

6. Applied rewrites 87.8%

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

7. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 84.0

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

9. Applied rewrites 84.0%

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

10. Final simplification 84.0%

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

Developer Target 1: 97.9% 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 2024284 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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