Quotient of products

Percentage Accurate: 85.8% → 97.2%

Time: 3.7s

Alternatives: 4

Speedup: 0.6×

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: 85.8% 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: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \cdot b2\_m \leq 10^{-90}:\\ \;\;\;\;a2\_m \cdot \frac{\frac{a1\_m}{b2\_m}}{b1\_m}\\ \mathbf{elif}\;b1\_m \cdot b2\_m \leq 2 \cdot 10^{+240}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b1\_m \cdot b2\_m}\\ \mathbf{else}:\\ \;\;\;\;a1\_m \cdot \frac{\frac{a2\_m}{b1\_m}}{b2\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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.
(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 (<= (* b1_m b2_m) 1e-90)
       (* a2_m (/ (/ a1_m b2_m) b1_m))
       (if (<= (* b1_m b2_m) 2e+240)
         (* 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);
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) <= 1e-90) {
		tmp = a2_m * ((a1_m / b2_m) / b1_m);
	} else if ((b1_m * b2_m) <= 2e+240) {
		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.
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) <= 1d-90) then
        tmp = a2_m * ((a1_m / b2_m) / b1_m)
    else if ((b1_m * b2_m) <= 2d+240) 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;
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) <= 1e-90) {
		tmp = a2_m * ((a1_m / b2_m) / b1_m);
	} else if ((b1_m * b2_m) <= 2e+240) {
		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])
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) <= 1e-90:
		tmp = a2_m * ((a1_m / b2_m) / b1_m)
	elif (b1_m * b2_m) <= 2e+240:
		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])
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) <= 1e-90)
		tmp = Float64(a2_m * Float64(Float64(a1_m / b2_m) / b1_m));
	elseif (Float64(b1_m * b2_m) <= 2e+240)
		tmp = Float64(a1_m * Float64(a2_m / Float64(b1_m * b2_m)));
	else
		tmp = Float64(a1_m * Float64(Float64(a2_m / 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])){:}
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) <= 1e-90)
		tmp = a2_m * ((a1_m / b2_m) / b1_m);
	elseif ((b1_m * b2_m) <= 2e+240)
		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.
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[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 1e-90], N[(a2$95$m * N[(N[(a1$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b1$95$m * b2$95$m), $MachinePrecision], 2e+240], N[(a1$95$m * N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1$95$m * N[(N[(a2$95$m / b1$95$m), $MachinePrecision] / b2$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < 9.99999999999999995e-91

   1. Initial program 84.7%

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

      1. associate-/l* 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in a1 around 0 84.7%

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

      1. *-commutative 84.7%

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

      2. times-frac 81.1%

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

      3. associate-*l/ 88.7%

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

      4. associate-*r/ 86.6%

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

   7. Simplified 86.6%

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

   if 9.99999999999999995e-91 < (*.f64 b1 b2) < 2.00000000000000003e240

   1. Initial program 96.1%

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

      1. associate-/l* 92.8%

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

   3. Simplified 92.8%

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

   if 2.00000000000000003e240 < (*.f64 b1 b2)

   1. Initial program 74.1%

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

      1. associate-/l* 74.3%

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

      2. associate-/r* 91.4%

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

   3. Simplified 91.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq 10^{-90}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;b1\_m \leq 6.8 \cdot 10^{-44}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b1\_m \cdot b2\_m}\\ \mathbf{else}:\\ \;\;\;\;a1\_m \cdot \frac{\frac{a2\_m}{b1\_m}}{b2\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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.
(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 (<= b1_m 6.8e-44)
       (* 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);
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 <= 6.8e-44) {
		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.
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 <= 6.8d-44) 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;
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 <= 6.8e-44) {
		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])
def code(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m):
	tmp = 0
	if b1_m <= 6.8e-44:
		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])
function code(b2_s, b1_s, a2_s, a1_s, a1_m, a2_m, b1_m, b2_m)
	tmp = 0.0
	if (b1_m <= 6.8e-44)
		tmp = Float64(a1_m * Float64(a2_m / Float64(b1_m * b2_m)));
	else
		tmp = Float64(a1_m * Float64(Float64(a2_m / 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])){:}
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 <= 6.8e-44)
		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.
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[LessEqual[b1$95$m, 6.8e-44], N[(a1$95$m * N[(a2$95$m / N[(b1$95$m * b2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1$95$m * N[(N[(a2$95$m / b1$95$m), $MachinePrecision] / b2$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b1 < 6.80000000000000033e-44

   1. Initial program 85.6%

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

      1. associate-/l* 81.3%

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

   3. Simplified 81.3%

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

   if 6.80000000000000033e-44 < b1

   1. Initial program 87.2%

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

      1. associate-/l* 77.6%

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

      2. associate-/r* 78.4%

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

   3. Simplified 78.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \leq 6.8 \cdot 10^{-44}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 1.0× speedup

Math

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

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.
(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);
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.
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;
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])
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])
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])){:}
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.
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 86.1%

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

   1. associate-/l* 80.2%

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

3. Simplified 80.2%

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

5. Final simplification 80.2%

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

Alternative 4: 85.7% accurate, 1.0× speedup

Math

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

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.
(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);
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.
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;
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])
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])
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])){:}
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.
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 86.1%

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

   1. times-frac 84.8%

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

3. Simplified 84.8%

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

5. Final simplification 84.8%

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

Reproduce

herbie shell --seed 2024059 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64
  (/ (* a1 a2) (* b1 b2)))