Quotient of products

Percentage Accurate: 85.3% → 97.5%

Time: 7.5s

Alternatives: 7

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: 85.3% 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: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;b2\_m \cdot b1\_m \leq 0:\\ \;\;\;\;\frac{a2\_m}{b1\_m} \cdot \frac{a1\_m}{b2\_m}\\ \mathbf{elif}\;b2\_m \cdot b1\_m \leq 10^{-77}:\\ \;\;\;\;a2\_m \cdot \frac{a1\_m}{b2\_m \cdot b1\_m}\\ \mathbf{else}:\\ \;\;\;\;a1\_m \cdot \frac{\frac{a2\_m}{b2\_m}}{b1\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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 (<= (* b2_m b1_m) 0.0)
       (* (/ a2_m b1_m) (/ a1_m b2_m))
       (if (<= (* b2_m b1_m) 1e-77)
         (* a2_m (/ a1_m (* b2_m b1_m)))
         (* a1_m (/ (/ a2_m b2_m) b1_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 ((b2_m * b1_m) <= 0.0) {
		tmp = (a2_m / b1_m) * (a1_m / b2_m);
	} else if ((b2_m * b1_m) <= 1e-77) {
		tmp = a2_m * (a1_m / (b2_m * b1_m));
	} else {
		tmp = a1_m * ((a2_m / b2_m) / b1_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 ((b2_m * b1_m) <= 0.0d0) then
        tmp = (a2_m / b1_m) * (a1_m / b2_m)
    else if ((b2_m * b1_m) <= 1d-77) then
        tmp = a2_m * (a1_m / (b2_m * b1_m))
    else
        tmp = a1_m * ((a2_m / b2_m) / b1_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 ((b2_m * b1_m) <= 0.0) {
		tmp = (a2_m / b1_m) * (a1_m / b2_m);
	} else if ((b2_m * b1_m) <= 1e-77) {
		tmp = a2_m * (a1_m / (b2_m * b1_m));
	} else {
		tmp = a1_m * ((a2_m / b2_m) / b1_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 (b2_m * b1_m) <= 0.0:
		tmp = (a2_m / b1_m) * (a1_m / b2_m)
	elif (b2_m * b1_m) <= 1e-77:
		tmp = a2_m * (a1_m / (b2_m * b1_m))
	else:
		tmp = a1_m * ((a2_m / b2_m) / b1_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(b2_m * b1_m) <= 0.0)
		tmp = Float64(Float64(a2_m / b1_m) * Float64(a1_m / b2_m));
	elseif (Float64(b2_m * b1_m) <= 1e-77)
		tmp = Float64(a2_m * Float64(a1_m / Float64(b2_m * b1_m)));
	else
		tmp = Float64(a1_m * Float64(Float64(a2_m / b2_m) / b1_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 ((b2_m * b1_m) <= 0.0)
		tmp = (a2_m / b1_m) * (a1_m / b2_m);
	elseif ((b2_m * b1_m) <= 1e-77)
		tmp = a2_m * (a1_m / (b2_m * b1_m));
	else
		tmp = a1_m * ((a2_m / b2_m) / b1_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[LessEqual[N[(b2$95$m * b1$95$m), $MachinePrecision], 0.0], N[(N[(a2$95$m / b1$95$m), $MachinePrecision] * N[(a1$95$m / b2$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b2$95$m * b1$95$m), $MachinePrecision], 1e-77], N[(a2$95$m * N[(a1$95$m / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1$95$m * N[(N[(a2$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < 0.0

   1. Initial program 84.5%

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

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 85.8

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

   4. Applied egg-rr 85.8%

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

   if 0.0 < (*.f64 b1 b2) < 9.9999999999999993e-78

   1. Initial program 83.5%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 84.8

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

   4. Applied egg-rr 84.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2 + 0} \cdot a2} \]

      8. +-rgt-identity N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-rgt-identity N/A

          \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      11. *-commutative N/A

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1} + 0} \cdot a2 \]

      12. accelerator-lowering-fma.f64 92.0

          \[\leadsto \frac{a1}{\color{blue}{\mathsf{fma}\left(b2, b1, 0\right)}} \cdot a2 \]

   6. Applied egg-rr 92.0%

      \[\leadsto \color{blue}{\frac{a1}{\mathsf{fma}\left(b2, b1, 0\right)} \cdot a2} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 92.0

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

   8. Applied egg-rr 92.0%

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

   if 9.9999999999999993e-78 < (*.f64 b1 b2)

   1. Initial program 86.7%

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

   3. Taylor expanded in a1 around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 84.8

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

   7. Applied egg-rr 84.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b2 \cdot b1 \leq 0:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b2 \cdot b1 \leq 10^{-77}:\\ \;\;\;\;a2 \cdot \frac{a1}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a2\_m \cdot a1\_m}{b2\_m \cdot b1\_m} \leq 10^{-15}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b2\_m \cdot b1\_m}\\ \mathbf{else}:\\ \;\;\;\;a2\_m \cdot \left(a1\_m \cdot \frac{1}{b2\_m \cdot b1\_m}\right)\\ \end{array}\right)\right)\right) \end{array} \]

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 (<= (/ (* a2_m a1_m) (* b2_m b1_m)) 1e-15)
       (* a1_m (/ a2_m (* b2_m b1_m)))
       (* a2_m (* a1_m (/ 1.0 (* b2_m b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 1e-15) {
		tmp = a1_m * (a2_m / (b2_m * b1_m));
	} else {
		tmp = a2_m * (a1_m * (1.0 / (b2_m * b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 1d-15) then
        tmp = a1_m * (a2_m / (b2_m * b1_m))
    else
        tmp = a2_m * (a1_m * (1.0d0 / (b2_m * b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 1e-15) {
		tmp = a1_m * (a2_m / (b2_m * b1_m));
	} else {
		tmp = a2_m * (a1_m * (1.0 / (b2_m * b1_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 ((a2_m * a1_m) / (b2_m * b1_m)) <= 1e-15:
		tmp = a1_m * (a2_m / (b2_m * b1_m))
	else:
		tmp = a2_m * (a1_m * (1.0 / (b2_m * b1_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(Float64(a2_m * a1_m) / Float64(b2_m * b1_m)) <= 1e-15)
		tmp = Float64(a1_m * Float64(a2_m / Float64(b2_m * b1_m)));
	else
		tmp = Float64(a2_m * Float64(a1_m * Float64(1.0 / Float64(b2_m * b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 1e-15)
		tmp = a1_m * (a2_m / (b2_m * b1_m));
	else
		tmp = a2_m * (a1_m * (1.0 / (b2_m * b1_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[LessEqual[N[(N[(a2$95$m * a1$95$m), $MachinePrecision] / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision], 1e-15], N[(a1$95$m * N[(a2$95$m / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2$95$m * N[(a1$95$m * N[(1.0 / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.0000000000000001e-15

   1. Initial program 88.0%

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

   3. Taylor expanded in a1 around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 90.8

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

   5. Simplified 90.8%

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

   if 1.0000000000000001e-15 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 77.5%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 85.4

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

   4. Applied egg-rr 85.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2 + 0} \cdot a2} \]

      8. +-rgt-identity N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-rgt-identity N/A

          \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      11. *-commutative N/A

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1} + 0} \cdot a2 \]

      12. accelerator-lowering-fma.f64 77.6

          \[\leadsto \frac{a1}{\color{blue}{\mathsf{fma}\left(b2, b1, 0\right)}} \cdot a2 \]

   6. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\frac{a1}{\mathsf{fma}\left(b2, b1, 0\right)} \cdot a2} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 77.6

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

   8. Applied egg-rr 77.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{b2 \cdot b1} \cdot a1\right)} \cdot a2 \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{b2 \cdot b1} \cdot a1\right)} \cdot a2 \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{b2 \cdot b1}} \cdot a1\right) \cdot a2 \]

      5. *-lowering-*.f64 77.5

         \[\leadsto \left(\frac{1}{\color{blue}{b2 \cdot b1}} \cdot a1\right) \cdot a2 \]

   10. Applied egg-rr 77.5%

       \[\leadsto \color{blue}{\left(\frac{1}{b2 \cdot b1} \cdot a1\right)} \cdot a2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a2 \cdot a1}{b2 \cdot b1} \leq 10^{-15}:\\ \;\;\;\;a1 \cdot \frac{a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a1 \cdot \frac{1}{b2 \cdot b1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a2\_m \cdot a1\_m}{b2\_m \cdot b1\_m} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;a1\_m \cdot \frac{a2\_m}{b2\_m \cdot b1\_m}\\ \mathbf{else}:\\ \;\;\;\;a2\_m \cdot \frac{a1\_m}{b2\_m \cdot b1\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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 (<= (/ (* a2_m a1_m) (* b2_m b1_m)) 2e-26)
       (* a1_m (/ a2_m (* b2_m b1_m)))
       (* a2_m (/ a1_m (* b2_m b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 2e-26) {
		tmp = a1_m * (a2_m / (b2_m * b1_m));
	} else {
		tmp = a2_m * (a1_m / (b2_m * b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 2d-26) then
        tmp = a1_m * (a2_m / (b2_m * b1_m))
    else
        tmp = a2_m * (a1_m / (b2_m * b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 2e-26) {
		tmp = a1_m * (a2_m / (b2_m * b1_m));
	} else {
		tmp = a2_m * (a1_m / (b2_m * b1_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 ((a2_m * a1_m) / (b2_m * b1_m)) <= 2e-26:
		tmp = a1_m * (a2_m / (b2_m * b1_m))
	else:
		tmp = a2_m * (a1_m / (b2_m * b1_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(Float64(a2_m * a1_m) / Float64(b2_m * b1_m)) <= 2e-26)
		tmp = Float64(a1_m * Float64(a2_m / Float64(b2_m * b1_m)));
	else
		tmp = Float64(a2_m * Float64(a1_m / Float64(b2_m * b1_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 (((a2_m * a1_m) / (b2_m * b1_m)) <= 2e-26)
		tmp = a1_m * (a2_m / (b2_m * b1_m));
	else
		tmp = a2_m * (a1_m / (b2_m * b1_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[LessEqual[N[(N[(a2$95$m * a1$95$m), $MachinePrecision] / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision], 2e-26], N[(a1$95$m * N[(a2$95$m / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2$95$m * N[(a1$95$m / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.0000000000000001e-26

   1. Initial program 87.8%

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

   3. Taylor expanded in a1 around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 90.7

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

   5. Simplified 90.7%

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

   if 2.0000000000000001e-26 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 78.4%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 85.9

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

   4. Applied egg-rr 85.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2 + 0} \cdot a2} \]

      8. +-rgt-identity N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-rgt-identity N/A

          \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      11. *-commutative N/A

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1} + 0} \cdot a2 \]

      12. accelerator-lowering-fma.f64 78.4

          \[\leadsto \frac{a1}{\color{blue}{\mathsf{fma}\left(b2, b1, 0\right)}} \cdot a2 \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\frac{a1}{\mathsf{fma}\left(b2, b1, 0\right)} \cdot a2} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 78.4

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

   8. Applied egg-rr 78.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a2 \cdot a1}{b2 \cdot b1} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;a1 \cdot \frac{a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b2 \cdot b1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a1\_m = \left|a1\right| \\ a1\_s = \mathsf{copysign}\left(1, a1\right) \\ a2\_m = \left|a2\right| \\ a2\_s = \mathsf{copysign}\left(1, a2\right) \\ b1\_m = \left|b1\right| \\ b1\_s = \mathsf{copysign}\left(1, b1\right) \\ b2\_m = \left|b2\right| \\ b2\_s = \mathsf{copysign}\left(1, b2\right) \\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\\\ [a1_m, a2_m, b1_m, b2_m] = \mathsf{sort}([a1_m, a2_m, b1_m, b2_m])\\ \\ b2\_s \cdot \left(b1\_s \cdot \left(a2\_s \cdot \left(a1\_s \cdot \begin{array}{l} \mathbf{if}\;b2\_m \cdot b1\_m \leq 10^{-77}:\\ \;\;\;\;a2\_m \cdot \frac{a1\_m}{b2\_m \cdot b1\_m}\\ \mathbf{else}:\\ \;\;\;\;a1\_m \cdot \frac{\frac{a2\_m}{b2\_m}}{b1\_m}\\ \end{array}\right)\right)\right) \end{array} \]

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 (<= (* b2_m b1_m) 1e-77)
       (* a2_m (/ a1_m (* b2_m b1_m)))
       (* a1_m (/ (/ a2_m b2_m) b1_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 ((b2_m * b1_m) <= 1e-77) {
		tmp = a2_m * (a1_m / (b2_m * b1_m));
	} else {
		tmp = a1_m * ((a2_m / b2_m) / b1_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 ((b2_m * b1_m) <= 1d-77) then
        tmp = a2_m * (a1_m / (b2_m * b1_m))
    else
        tmp = a1_m * ((a2_m / b2_m) / b1_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 ((b2_m * b1_m) <= 1e-77) {
		tmp = a2_m * (a1_m / (b2_m * b1_m));
	} else {
		tmp = a1_m * ((a2_m / b2_m) / b1_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 (b2_m * b1_m) <= 1e-77:
		tmp = a2_m * (a1_m / (b2_m * b1_m))
	else:
		tmp = a1_m * ((a2_m / b2_m) / b1_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(b2_m * b1_m) <= 1e-77)
		tmp = Float64(a2_m * Float64(a1_m / Float64(b2_m * b1_m)));
	else
		tmp = Float64(a1_m * Float64(Float64(a2_m / b2_m) / b1_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 ((b2_m * b1_m) <= 1e-77)
		tmp = a2_m * (a1_m / (b2_m * b1_m));
	else
		tmp = a1_m * ((a2_m / b2_m) / b1_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[LessEqual[N[(b2$95$m * b1$95$m), $MachinePrecision], 1e-77], N[(a2$95$m * N[(a1$95$m / N[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a1$95$m * N[(N[(a2$95$m / b2$95$m), $MachinePrecision] / b1$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b1 b2) < 9.9999999999999993e-78

   1. Initial program 84.3%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 86.0

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

   4. Applied egg-rr 86.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a1}{b1 \cdot b2 + 0} \cdot a2} \]

      8. +-rgt-identity N/A

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

      9. /-lowering-/.f64 N/A

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

      10. +-rgt-identity N/A

          \[\leadsto \frac{a1}{\color{blue}{b1 \cdot b2 + 0}} \cdot a2 \]

      11. *-commutative N/A

          \[\leadsto \frac{a1}{\color{blue}{b2 \cdot b1} + 0} \cdot a2 \]

      12. accelerator-lowering-fma.f64 84.6

          \[\leadsto \frac{a1}{\color{blue}{\mathsf{fma}\left(b2, b1, 0\right)}} \cdot a2 \]

   6. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{\frac{a1}{\mathsf{fma}\left(b2, b1, 0\right)} \cdot a2} \]
   7. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 84.6

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

   8. Applied egg-rr 84.6%

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

   if 9.9999999999999993e-78 < (*.f64 b1 b2)

   1. Initial program 86.7%

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

   3. Taylor expanded in a1 around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 84.8

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

   7. Applied egg-rr 84.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b2 \cdot b1 \leq 10^{-77}:\\ \;\;\;\;a2 \cdot \frac{a1}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.6× speedup

Math

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

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 (/ (/ a2_m b2_m) (/ b1_m a1_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 * ((a2_m / b2_m) / (b1_m / a1_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 * ((a2_m / b2_m) / (b1_m / a1_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 * ((a2_m / b2_m) / (b1_m / a1_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 * ((a2_m / b2_m) / (b1_m / a1_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(a2_m / b2_m) / Float64(b1_m / a1_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 * ((a2_m / b2_m) / (b1_m / a1_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[(a2$95$m / b2$95$m), $MachinePrecision] / N[(b1$95$m / a1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. clear-num N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 85.5

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

4. Applied egg-rr 85.5%

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

Alternative 6: 97.9% accurate, 0.8× speedup

Math

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

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 (* (/ a2_m b2_m) (/ a1_m b1_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 * ((a2_m / b2_m) * (a1_m / b1_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 * ((a2_m / b2_m) * (a1_m / b1_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 * ((a2_m / b2_m) * (a1_m / b1_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 * ((a2_m / b2_m) * (a1_m / b1_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(a2_m / b2_m) * Float64(a1_m / b1_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 * ((a2_m / b2_m) * (a1_m / b1_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[(a2$95$m / b2$95$m), $MachinePrecision] * N[(a1$95$m / b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 85.5

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

4. Applied egg-rr 85.5%

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

Alternative 7: 85.4% accurate, 1.0× speedup

Math

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

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 (* b2_m b1_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 / (b2_m * b1_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 / (b2_m * b1_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 / (b2_m * b1_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 / (b2_m * b1_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(b2_m * b1_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 / (b2_m * b1_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[(b2$95$m * b1$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in a1 around 0

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 87.4

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

5. Simplified 87.4%

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

6. Final simplification 87.4%

   \[\leadsto a1 \cdot \frac{a2}{b2 \cdot b1} \]
7. 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 2024198 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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