Quotient of products

Percentage Accurate: 85.9% → 97.9%

Time: 3.4s

Alternatives: 4

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: 85.9% 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, 1.0× speedup

Math

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

FPCore

a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
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 (b2_s b1_s a2_s a1_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (* b2_s (* b1_s (* a2_s (* a1_s (/ (/ a1_m b1_m) (/ b2_m a2_m)))))))

C

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

Fortran

a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: a1_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 = b2_s * (b1_s * (a2_s * (a1_s * ((a1_m / b1_m) / (b2_m / a2_m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
b2\_m = abs(b2);
b2\_s = sign(b2) * 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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
	tmp = b2_s * (b1_s * (a2_s * (a1_s * ((a1_m / b1_m) / (b2_m / a2_m)))));
end

Wolfram

a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, 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]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, 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[b2$95$s_, b1$95$s_, a2$95$s_, a1$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(b2$95$s * N[(b1$95$s * N[(a2$95$s * N[(a1$95$s * N[(N[(a1$95$m / b1$95$m), $MachinePrecision] / N[(b2$95$m / a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 88.3%

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

3. Simplified 88.3%

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

   1. associate-*r/ 88.0%

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

   2. frac-times 89.6%

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

   3. clear-num 89.1%

      \[\leadsto \frac{a1}{b1} \cdot \color{blue}{\frac{1}{\frac{b2}{a2}}} \]

   4. un-div-inv 89.6%

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

6. Applied egg-rr 89.6%

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

7. Final simplification 89.6%

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

Alternative 2: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 5 \cdot 10^{-167} \lor \neg \left(b1\_m \cdot b2\_m \leq 4 \cdot 10^{+158}\right):\\ \;\;\;\;a1\_m \cdot \frac{\frac{a2\_m}{b1\_m}}{b2\_m}\\ \mathbf{else}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b1\_m \cdot b2\_m}\\ \end{array}\right)\right)\right) \end{array} \]

FPCore

a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
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 (b2_s b1_s a2_s a1_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (*
  b2_s
  (*
   b1_s
   (*
    a2_s
    (*
     a1_s
     (if (or (<= (* b1_m b2_m) 5e-167) (not (<= (* b1_m b2_m) 4e+158)))
       (* a1_m (/ (/ a2_m b1_m) b2_m))
       (* a1_m (/ a2_m (* b1_m b2_m)))))))))

C

a1\_m = fabs(a1);
a1\_s = copysign(1.0, a1);
a2\_m = fabs(a2);
a2\_s = copysign(1.0, a2);
b1\_m = fabs(b1);
b1\_s = copysign(1.0, b1);
b2\_m = fabs(b2);
b2\_s = copysign(1.0, b2);
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 b2_s, double b1_s, double a2_s, double a1_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if (((b1_m * b2_m) <= 5e-167) || !((b1_m * b2_m) <= 4e+158)) {
		tmp = a1_m * ((a2_m / b1_m) / b2_m);
	} else {
		tmp = a1_m * (a2_m / (b1_m * b2_m));
	}
	return b2_s * (b1_s * (a2_s * (a1_s * tmp)));
}

Fortran

a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: a1_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-167) .or. (.not. ((b1_m * b2_m) <= 4d+158))) then
        tmp = a1_m * ((a2_m / b1_m) / b2_m)
    else
        tmp = a1_m * (a2_m / (b1_m * b2_m))
    end if
    code = b2_s * (b1_s * (a2_s * (a1_s * tmp)))
end function

Java

a1\_m = Math.abs(a1);
a1\_s = Math.copySign(1.0, a1);
a2\_m = Math.abs(a2);
a2\_s = Math.copySign(1.0, a2);
b1\_m = Math.abs(b1);
b1\_s = Math.copySign(1.0, b1);
b2\_m = Math.abs(b2);
b2\_s = Math.copySign(1.0, b2);
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 b2_s, double b1_s, double a2_s, double a1_s, double a1_m, double a2_m, double b1_m, double b2_m) {
	double tmp;
	if (((b1_m * b2_m) <= 5e-167) || !((b1_m * b2_m) <= 4e+158)) {
		tmp = a1_m * ((a2_m / b1_m) / b2_m);
	} else {
		tmp = a1_m * (a2_m / (b1_m * b2_m));
	}
	return b2_s * (b1_s * (a2_s * (a1_s * tmp)));
}

Python

a1\_m = math.fabs(a1)
a1\_s = math.copysign(1.0, a1)
a2\_m = math.fabs(a2)
a2\_s = math.copysign(1.0, a2)
b1\_m = math.fabs(b1)
b1\_s = math.copysign(1.0, b1)
b2\_m = math.fabs(b2)
b2\_s = math.copysign(1.0, b2)
[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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m):
	tmp = 0
	if ((b1_m * b2_m) <= 5e-167) or not ((b1_m * b2_m) <= 4e+158):
		tmp = a1_m * ((a2_m / b1_m) / b2_m)
	else:
		tmp = a1_m * (a2_m / (b1_m * b2_m))
	return b2_s * (b1_s * (a2_s * (a1_s * tmp)))

Julia

a1\_m = abs(a1)
a1\_s = copysign(1.0, a1)
a2\_m = abs(a2)
a2\_s = copysign(1.0, a2)
b1\_m = abs(b1)
b1\_s = copysign(1.0, b1)
b2\_m = abs(b2)
b2\_s = copysign(1.0, b2)
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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0
	if ((Float64(b1_m * b2_m) <= 5e-167) || !(Float64(b1_m * b2_m) <= 4e+158))
		tmp = Float64(a1_m * Float64(Float64(a2_m / b1_m) / b2_m));
	else
		tmp = Float64(a1_m * Float64(a2_m / Float64(b1_m * b2_m)));
	end
	return Float64(b2_s * Float64(b1_s * Float64(a2_s * Float64(a1_s * tmp))))
end

MATLAB

a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
b2\_m = abs(b2);
b2\_s = sign(b2) * 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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0;
	if (((b1_m * b2_m) <= 5e-167) || ~(((b1_m * b2_m) <= 4e+158)))
		tmp = a1_m * ((a2_m / b1_m) / b2_m);
	else
		tmp = a1_m * (a2_m / (b1_m * b2_m));
	end
	tmp_2 = b2_s * (b1_s * (a2_s * (a1_s * tmp)));
end

Wolfram

a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, 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]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, 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[b2$95$s_, b1$95$s_, a2$95$s_, a1$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(b2$95$s * N[(b1$95$s * N[(a2$95$s * N[(a1$95$s * If[Or[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 5e-167], N[Not[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 4e+158]], $MachinePrecision]], N[(a1$95$m * N[(N[(a2$95$m / b1$95$m), $MachinePrecision] / b2$95$m), $MachinePrecision]), $MachinePrecision], N[(a1$95$m * N[(a2$95$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 b1 b2) < 5.0000000000000002e-167 or 3.99999999999999981e158 < (*.f64 b1 b2)

   1. Initial program 85.4%

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

      1. associate-/l* 86.6%

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

      2. associate-/r* 88.6%

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

   3. Simplified 88.6%

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

   if 5.0000000000000002e-167 < (*.f64 b1 b2) < 3.99999999999999981e158

   1. Initial program 97.9%

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

      1. associate-/l* 94.5%

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

   3. Simplified 94.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq 5 \cdot 10^{-167} \lor \neg \left(b1 \cdot b2 \leq 4 \cdot 10^{+158}\right):\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 1.0× speedup

Math

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

FPCore

a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
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 (b2_s b1_s a2_s a1_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (* b2_s (* b1_s (* a2_s (* a1_s (* a1_m (/ a2_m (* b1_m b2_m))))))))

C

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

Fortran

a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: a1_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 = b2_s * (b1_s * (a2_s * (a1_s * (a1_m * (a2_m / (b1_m * b2_m))))))
end function

Java

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

Python

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

Julia

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

MATLAB

a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
b2\_m = abs(b2);
b2\_s = sign(b2) * 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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
	tmp = b2_s * (b1_s * (a2_s * (a1_s * (a1_m * (a2_m / (b1_m * b2_m))))));
end

Wolfram

a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, 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]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, 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[b2$95$s_, b1$95$s_, a2$95$s_, a1$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(b2$95$s * N[(b1$95$s * N[(a2$95$s * N[(a1$95$s * N[(a1$95$m * N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 88.3%

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

3. Simplified 88.3%

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

5. Final simplification 88.3%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

a1\_m = (fabs.f64 a1)
a1\_s = (copysign.f64 #s(literal 1 binary64) a1)
a2\_m = (fabs.f64 a2)
a2\_s = (copysign.f64 #s(literal 1 binary64) a2)
b1\_m = (fabs.f64 b1)
b1\_s = (copysign.f64 #s(literal 1 binary64) b1)
b2\_m = (fabs.f64 b2)
b2\_s = (copysign.f64 #s(literal 1 binary64) b2)
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 (b2_s b1_s a2_s a1_s a1_m a2_m b1_m b2_m)
 :precision binary64
 (* b2_s (* b1_s (* a2_s (* a1_s (* (/ a1_m b1_m) (/ a2_m b2_m)))))))

C

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

Fortran

a1\_m = abs(a1)
a1\_s = copysign(1.0d0, a1)
a2\_m = abs(a2)
a2\_s = copysign(1.0d0, a2)
b1\_m = abs(b1)
b1\_s = copysign(1.0d0, b1)
b2\_m = abs(b2)
b2\_s = copysign(1.0d0, b2)
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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
    real(8), intent (in) :: b2_s
    real(8), intent (in) :: b1_s
    real(8), intent (in) :: a2_s
    real(8), intent (in) :: a1_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 = b2_s * (b1_s * (a2_s * (a1_s * ((a1_m / b1_m) * (a2_m / b2_m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

a1\_m = abs(a1);
a1\_s = sign(a1) * abs(1.0);
a2\_m = abs(a2);
a2\_s = sign(a2) * abs(1.0);
b1\_m = abs(b1);
b1\_s = sign(b1) * abs(1.0);
b2\_m = abs(b2);
b2\_s = sign(b2) * 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(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
	tmp = b2_s * (b1_s * (a2_s * (a1_s * ((a1_m / b1_m) * (a2_m / b2_m)))));
end

Wolfram

a1\_m = N[Abs[a1], $MachinePrecision]
a1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a1]}, 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]
b1\_m = N[Abs[b1], $MachinePrecision]
b1\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
b2\_m = N[Abs[b2], $MachinePrecision]
b2\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[b2]}, 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[b2$95$s_, b1$95$s_, a2$95$s_, a1$95$s_, a1$95$m_, a2$95$m_, b1$95$m_, b2$95$m_] := N[(b2$95$s * N[(b1$95$s * N[(a2$95$s * N[(a1$95$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 88.0%

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

   1. times-frac 89.6%

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

3. Simplified 89.6%

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

5. Final simplification 89.6%

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

Developer target: 97.9% accurate, 1.0× 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 2024066 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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