Quotient of products

Percentage Accurate: 86.6% → 96.8%

Time: 3.2s

Alternatives: 5

Speedup: 0.2×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a2}{b1 \cdot b2} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 * a2) / (b1 * b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 * a2) / Float64(b1 * b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 * a2) / (b1 * b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]

Initial Program: 86.6% 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: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ [b1, b2] = \mathsf{sort}([b1, b2])\\ \\ \begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq 10^{+275}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -5e+270)
     (* a2 (/ (/ a1 b2) b1))
     (if (<= t_0 -2e-309)
       t_0
       (if (<= t_0 0.0)
         (* (/ a1 b1) (/ a2 b2))
         (if (<= t_0 1e+275) t_0 (/ (/ a1 b2) (/ b1 a2))))))))

C

assert(a1 < a2);
assert(b1 < b2);
double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -5e+270) {
		tmp = a2 * ((a1 / b2) / b1);
	} else if (t_0 <= -2e-309) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 1e+275) {
		tmp = t_0;
	} else {
		tmp = (a1 / b2) / (b1 / a2);
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a1 * a2) / (b1 * b2)
    if (t_0 <= (-5d+270)) then
        tmp = a2 * ((a1 / b2) / b1)
    else if (t_0 <= (-2d-309)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a1 / b1) * (a2 / b2)
    else if (t_0 <= 1d+275) then
        tmp = t_0
    else
        tmp = (a1 / b2) / (b1 / a2)
    end if
    code = tmp
end function

Java

assert a1 < a2;
assert b1 < b2;
public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -5e+270) {
		tmp = a2 * ((a1 / b2) / b1);
	} else if (t_0 <= -2e-309) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 1e+275) {
		tmp = t_0;
	} else {
		tmp = (a1 / b2) / (b1 / a2);
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
[b1, b2] = sort([b1, b2])
def code(a1, a2, b1, b2):
	t_0 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_0 <= -5e+270:
		tmp = a2 * ((a1 / b2) / b1)
	elif t_0 <= -2e-309:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a1 / b1) * (a2 / b2)
	elif t_0 <= 1e+275:
		tmp = t_0
	else:
		tmp = (a1 / b2) / (b1 / a2)
	return tmp

Julia

a1, a2 = sort([a1, a2])
b1, b2 = sort([b1, b2])
function code(a1, a2, b1, b2)
	t_0 = Float64(Float64(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -5e+270)
		tmp = Float64(a2 * Float64(Float64(a1 / b2) / b1));
	elseif (t_0 <= -2e-309)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	elseif (t_0 <= 1e+275)
		tmp = t_0;
	else
		tmp = Float64(Float64(a1 / b2) / Float64(b1 / a2));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
b1, b2 = num2cell(sort([b1, b2])){:}
function tmp_2 = code(a1, a2, b1, b2)
	t_0 = (a1 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -5e+270)
		tmp = a2 * ((a1 / b2) / b1);
	elseif (t_0 <= -2e-309)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a1 / b1) * (a2 / b2);
	elseif (t_0 <= 1e+275)
		tmp = t_0;
	else
		tmp = (a1 / b2) / (b1 / a2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+270], N[(a2 * N[(N[(a1 / b2), $MachinePrecision] / b1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-309], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+275], t$95$0, N[(N[(a1 / b2), $MachinePrecision] / N[(b1 / a2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.99999999999999976e270

   1. Initial program 81.3%

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

      1. times-frac 95.8%

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

   3. Simplified 95.8%

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

      1. frac-times 81.3%

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

      2. *-commutative 81.3%

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

      3. frac-times 92.0%

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

      4. clear-num 92.0%

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

      5. un-div-inv 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in a1 around 0 81.3%

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

      1. associate-*l/ 87.6%

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

      2. *-commutative 87.6%

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

      3. associate-/r* 97.9%

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

   8. Simplified 97.9%

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

   if -4.99999999999999976e270 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9999999999999988e-309 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 9.9999999999999996e274

   1. Initial program 98.8%

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

   if -1.9999999999999988e-309 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 75.4%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   if 9.9999999999999996e274 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 59.1%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

      1. frac-times 59.1%

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

      2. *-commutative 59.1%

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

      3. frac-times 97.7%

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

      4. clear-num 97.8%

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

      5. un-div-inv 97.7%

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

   5. Applied egg-rr 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+275}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array} \]

Alternative 2: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ [b1, b2] = \mathsf{sort}([b1, b2])\\ \\ \begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq 10^{+275}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -5e+270)
     (* a2 (/ (/ a1 b2) b1))
     (if (<= t_0 -2e-309)
       t_0
       (if (<= t_0 0.0)
         (* (/ a1 b1) (/ a2 b2))
         (if (<= t_0 1e+275) t_0 (* (/ a1 b2) (/ a2 b1))))))))

C

assert(a1 < a2);
assert(b1 < b2);
double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -5e+270) {
		tmp = a2 * ((a1 / b2) / b1);
	} else if (t_0 <= -2e-309) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 1e+275) {
		tmp = t_0;
	} else {
		tmp = (a1 / b2) * (a2 / b1);
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a1 * a2) / (b1 * b2)
    if (t_0 <= (-5d+270)) then
        tmp = a2 * ((a1 / b2) / b1)
    else if (t_0 <= (-2d-309)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a1 / b1) * (a2 / b2)
    else if (t_0 <= 1d+275) then
        tmp = t_0
    else
        tmp = (a1 / b2) * (a2 / b1)
    end if
    code = tmp
end function

Java

assert a1 < a2;
assert b1 < b2;
public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -5e+270) {
		tmp = a2 * ((a1 / b2) / b1);
	} else if (t_0 <= -2e-309) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 1e+275) {
		tmp = t_0;
	} else {
		tmp = (a1 / b2) * (a2 / b1);
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
[b1, b2] = sort([b1, b2])
def code(a1, a2, b1, b2):
	t_0 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_0 <= -5e+270:
		tmp = a2 * ((a1 / b2) / b1)
	elif t_0 <= -2e-309:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a1 / b1) * (a2 / b2)
	elif t_0 <= 1e+275:
		tmp = t_0
	else:
		tmp = (a1 / b2) * (a2 / b1)
	return tmp

Julia

a1, a2 = sort([a1, a2])
b1, b2 = sort([b1, b2])
function code(a1, a2, b1, b2)
	t_0 = Float64(Float64(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -5e+270)
		tmp = Float64(a2 * Float64(Float64(a1 / b2) / b1));
	elseif (t_0 <= -2e-309)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	elseif (t_0 <= 1e+275)
		tmp = t_0;
	else
		tmp = Float64(Float64(a1 / b2) * Float64(a2 / b1));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
b1, b2 = num2cell(sort([b1, b2])){:}
function tmp_2 = code(a1, a2, b1, b2)
	t_0 = (a1 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -5e+270)
		tmp = a2 * ((a1 / b2) / b1);
	elseif (t_0 <= -2e-309)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a1 / b1) * (a2 / b2);
	elseif (t_0 <= 1e+275)
		tmp = t_0;
	else
		tmp = (a1 / b2) * (a2 / b1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+270], N[(a2 * N[(N[(a1 / b2), $MachinePrecision] / b1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-309], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+275], t$95$0, N[(N[(a1 / b2), $MachinePrecision] * N[(a2 / b1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.99999999999999976e270

   1. Initial program 81.3%

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

      1. times-frac 95.8%

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

   3. Simplified 95.8%

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

      1. frac-times 81.3%

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

      2. *-commutative 81.3%

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

      3. frac-times 92.0%

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

      4. clear-num 92.0%

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

      5. un-div-inv 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in a1 around 0 81.3%

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

      1. associate-*l/ 87.6%

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

      2. *-commutative 87.6%

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

      3. associate-/r* 97.9%

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

   8. Simplified 97.9%

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

   if -4.99999999999999976e270 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9999999999999988e-309 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 9.9999999999999996e274

   1. Initial program 98.8%

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

   if -1.9999999999999988e-309 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 75.4%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   if 9.9999999999999996e274 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 59.1%

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

      1. associate-/l* 62.2%

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

      2. *-commutative 62.2%

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

      3. associate-/l* 97.7%

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

   3. Simplified 97.7%

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

      1. associate-/r/ 97.7%

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

      2. *-commutative 97.7%

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

   5. Applied egg-rr 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+275}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \]

Alternative 3: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ [b1, b2] = \mathsf{sort}([b1, b2])\\ \\ \begin{array}{l} \mathbf{if}\;b2 \leq 3.4 \cdot 10^{+207}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= b2 3.4e+207) (* a2 (/ (/ a1 b2) b1)) (* (/ a1 b1) (/ a2 b2))))

C

assert(a1 < a2);
assert(b1 < b2);
double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (b2 <= 3.4e+207) {
		tmp = a2 * ((a1 / b2) / b1);
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (b2 <= 3.4d+207) then
        tmp = a2 * ((a1 / b2) / b1)
    else
        tmp = (a1 / b1) * (a2 / b2)
    end if
    code = tmp
end function

Java

assert a1 < a2;
assert b1 < b2;
public static double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (b2 <= 3.4e+207) {
		tmp = a2 * ((a1 / b2) / b1);
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
[b1, b2] = sort([b1, b2])
def code(a1, a2, b1, b2):
	tmp = 0
	if b2 <= 3.4e+207:
		tmp = a2 * ((a1 / b2) / b1)
	else:
		tmp = (a1 / b1) * (a2 / b2)
	return tmp

Julia

a1, a2 = sort([a1, a2])
b1, b2 = sort([b1, b2])
function code(a1, a2, b1, b2)
	tmp = 0.0
	if (b2 <= 3.4e+207)
		tmp = Float64(a2 * Float64(Float64(a1 / b2) / b1));
	else
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
b1, b2 = num2cell(sort([b1, b2])){:}
function tmp_2 = code(a1, a2, b1, b2)
	tmp = 0.0;
	if (b2 <= 3.4e+207)
		tmp = a2 * ((a1 / b2) / b1);
	else
		tmp = (a1 / b1) * (a2 / b2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
code[a1_, a2_, b1_, b2_] := If[LessEqual[b2, 3.4e+207], N[(a2 * N[(N[(a1 / b2), $MachinePrecision] / b1), $MachinePrecision]), $MachinePrecision], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b2 < 3.3999999999999998e207

   1. Initial program 84.1%

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

      1. times-frac 88.3%

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

   3. Simplified 88.3%

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

      1. frac-times 84.1%

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

      2. *-commutative 84.1%

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

      3. frac-times 90.1%

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

      4. clear-num 90.1%

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

      5. un-div-inv 90.5%

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

   5. Applied egg-rr 90.5%

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

   6. Taylor expanded in a1 around 0 84.1%

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

      1. associate-*l/ 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-/r* 89.1%

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

   8. Simplified 89.1%

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

   if 3.3999999999999998e207 < b2

   1. Initial program 78.6%

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

      1. times-frac 73.9%

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

   3. Simplified 73.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b2 \leq 3.4 \cdot 10^{+207}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 4: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ [b1, b2] = \mathsf{sort}([b1, b2])\\ \\ \begin{array}{l} \mathbf{if}\;b2 \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= b2 3.8e+195) (* (/ a1 b2) (/ a2 b1)) (* (/ a1 b1) (/ a2 b2))))

C

assert(a1 < a2);
assert(b1 < b2);
double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (b2 <= 3.8e+195) {
		tmp = (a1 / b2) * (a2 / b1);
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (b2 <= 3.8d+195) then
        tmp = (a1 / b2) * (a2 / b1)
    else
        tmp = (a1 / b1) * (a2 / b2)
    end if
    code = tmp
end function

Java

assert a1 < a2;
assert b1 < b2;
public static double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (b2 <= 3.8e+195) {
		tmp = (a1 / b2) * (a2 / b1);
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Python

[a1, a2] = sort([a1, a2])
[b1, b2] = sort([b1, b2])
def code(a1, a2, b1, b2):
	tmp = 0
	if b2 <= 3.8e+195:
		tmp = (a1 / b2) * (a2 / b1)
	else:
		tmp = (a1 / b1) * (a2 / b2)
	return tmp

Julia

a1, a2 = sort([a1, a2])
b1, b2 = sort([b1, b2])
function code(a1, a2, b1, b2)
	tmp = 0.0
	if (b2 <= 3.8e+195)
		tmp = Float64(Float64(a1 / b2) * Float64(a2 / b1));
	else
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	end
	return tmp
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
b1, b2 = num2cell(sort([b1, b2])){:}
function tmp_2 = code(a1, a2, b1, b2)
	tmp = 0.0;
	if (b2 <= 3.8e+195)
		tmp = (a1 / b2) * (a2 / b1);
	else
		tmp = (a1 / b1) * (a2 / b2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
code[a1_, a2_, b1_, b2_] := If[LessEqual[b2, 3.8e+195], N[(N[(a1 / b2), $MachinePrecision] * N[(a2 / b1), $MachinePrecision]), $MachinePrecision], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b2 < 3.8e195

   1. Initial program 83.8%

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

      1. associate-/l* 81.4%

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

      2. *-commutative 81.4%

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

      3. associate-/l* 87.8%

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

   3. Simplified 87.8%

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

      1. associate-/r/ 90.8%

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

      2. *-commutative 90.8%

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

   5. Applied egg-rr 90.8%

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

   if 3.8e195 < b2

   1. Initial program 82.4%

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

      1. times-frac 74.3%

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

   3. Simplified 74.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b2 \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a1, a2] = \mathsf{sort}([a1, a2])\\ [b1, b2] = \mathsf{sort}([b1, b2])\\ \\ a2 \cdot \frac{\frac{a1}{b2}}{b1} \end{array} \]

FPCore

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
(FPCore (a1 a2 b1 b2) :precision binary64 (* a2 (/ (/ a1 b2) b1)))

C

assert(a1 < a2);
assert(b1 < b2);
double code(double a1, double a2, double b1, double b2) {
	return a2 * ((a1 / b2) / b1);
}

Fortran

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
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 = a2 * ((a1 / b2) / b1)
end function

Java

assert a1 < a2;
assert b1 < b2;
public static double code(double a1, double a2, double b1, double b2) {
	return a2 * ((a1 / b2) / b1);
}

Python

[a1, a2] = sort([a1, a2])
[b1, b2] = sort([b1, b2])
def code(a1, a2, b1, b2):
	return a2 * ((a1 / b2) / b1)

Julia

a1, a2 = sort([a1, a2])
b1, b2 = sort([b1, b2])
function code(a1, a2, b1, b2)
	return Float64(a2 * Float64(Float64(a1 / b2) / b1))
end

MATLAB

a1, a2 = num2cell(sort([a1, a2])){:}
b1, b2 = num2cell(sort([b1, b2])){:}
function tmp = code(a1, a2, b1, b2)
	tmp = a2 * ((a1 / b2) / b1);
end

Wolfram

NOTE: a1 and a2 should be sorted in increasing order before calling this function.
NOTE: b1 and b2 should be sorted in increasing order before calling this function.
code[a1_, a2_, b1_, b2_] := N[(a2 * N[(N[(a1 / b2), $MachinePrecision] / b1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

   1. times-frac 87.3%

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

3. Simplified 87.3%

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

   1. frac-times 83.7%

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

   2. *-commutative 83.7%

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

   3. frac-times 90.6%

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

   4. clear-num 90.5%

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

   5. un-div-inv 91.0%

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

5. Applied egg-rr 91.0%

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

6. Taylor expanded in a1 around 0 83.7%

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

   1. associate-*l/ 84.8%

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

   2. *-commutative 84.8%

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

   3. associate-/r* 89.3%

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

8. Simplified 89.3%

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

9. Final simplification 89.3%

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

Developer target: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a1}{b1} \cdot \frac{a2}{b2} \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (* (/ a1 b1) (/ a2 b2)))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 / b1) * (a2 / b2);
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 / b1) * (a2 / b2)
end function

Java

public static double code(double a1, double a2, double b1, double b2) {
	return (a1 / b1) * (a2 / b2);
}

Python

def code(a1, a2, b1, b2):
	return (a1 / b1) * (a2 / b2)

Julia

function code(a1, a2, b1, b2)
	return Float64(Float64(a1 / b1) * Float64(a2 / b2))
end

MATLAB

function tmp = code(a1, a2, b1, b2)
	tmp = (a1 / b1) * (a2 / b2);
end

Wolfram

code[a1_, a2_, b1_, b2_] := N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023185 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

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