Quotient of products

Percentage Accurate: 85.5% → 92.7%

Time: 7.0s

Precision: binary64

Cost: 1489

• Unsound rule application detected in e-graph. Results may not be sound.

• 20.2% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-266} \lor \neg \left(b1 \cdot b2 \leq 5 \cdot 10^{+133}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (* (/ a1 b1) (/ a2 b2))))
   (if (<= (* b1 b2) -1e+198)
     t_0
     (if (<= (* b1 b2) -5e-113)
       (/ a2 (/ (* b1 b2) a1))
       (if (or (<= (* b1 b2) 5e-266) (not (<= (* b1 b2) 5e+133)))
         t_0
         (* a1 (/ a2 (* b1 b2))))))))

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 / b1) * (a2 / b2);
	double tmp;
	if ((b1 * b2) <= -1e+198) {
		tmp = t_0;
	} else if ((b1 * b2) <= -5e-113) {
		tmp = a2 / ((b1 * b2) / a1);
	} else if (((b1 * b2) <= 5e-266) || !((b1 * b2) <= 5e+133)) {
		tmp = t_0;
	} else {
		tmp = a1 * (a2 / (b1 * b2));
	}
	return tmp;
}

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

↓

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 / b1) * (a2 / b2)
    if ((b1 * b2) <= (-1d+198)) then
        tmp = t_0
    else if ((b1 * b2) <= (-5d-113)) then
        tmp = a2 / ((b1 * b2) / a1)
    else if (((b1 * b2) <= 5d-266) .or. (.not. ((b1 * b2) <= 5d+133))) then
        tmp = t_0
    else
        tmp = a1 * (a2 / (b1 * b2))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 / b1) * (a2 / b2);
	double tmp;
	if ((b1 * b2) <= -1e+198) {
		tmp = t_0;
	} else if ((b1 * b2) <= -5e-113) {
		tmp = a2 / ((b1 * b2) / a1);
	} else if (((b1 * b2) <= 5e-266) || !((b1 * b2) <= 5e+133)) {
		tmp = t_0;
	} else {
		tmp = a1 * (a2 / (b1 * b2));
	}
	return tmp;
}

Python

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

↓

def code(a1, a2, b1, b2):
	t_0 = (a1 / b1) * (a2 / b2)
	tmp = 0
	if (b1 * b2) <= -1e+198:
		tmp = t_0
	elif (b1 * b2) <= -5e-113:
		tmp = a2 / ((b1 * b2) / a1)
	elif ((b1 * b2) <= 5e-266) or not ((b1 * b2) <= 5e+133):
		tmp = t_0
	else:
		tmp = a1 * (a2 / (b1 * b2))
	return tmp

Julia

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

↓

function code(a1, a2, b1, b2)
	t_0 = Float64(Float64(a1 / b1) * Float64(a2 / b2))
	tmp = 0.0
	if (Float64(b1 * b2) <= -1e+198)
		tmp = t_0;
	elseif (Float64(b1 * b2) <= -5e-113)
		tmp = Float64(a2 / Float64(Float64(b1 * b2) / a1));
	elseif ((Float64(b1 * b2) <= 5e-266) || !(Float64(b1 * b2) <= 5e+133))
		tmp = t_0;
	else
		tmp = Float64(a1 * Float64(a2 / Float64(b1 * b2)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a1, a2, b1, b2)
	t_0 = (a1 / b1) * (a2 / b2);
	tmp = 0.0;
	if ((b1 * b2) <= -1e+198)
		tmp = t_0;
	elseif ((b1 * b2) <= -5e-113)
		tmp = a2 / ((b1 * b2) / a1);
	elseif (((b1 * b2) <= 5e-266) || ~(((b1 * b2) <= 5e+133)))
		tmp = t_0;
	else
		tmp = a1 * (a2 / (b1 * b2));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a1_, a2_, b1_, b2_] := Block[{t$95$0 = N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b1 * b2), $MachinePrecision], -1e+198], t$95$0, If[LessEqual[N[(b1 * b2), $MachinePrecision], -5e-113], N[(a2 / N[(N[(b1 * b2), $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(b1 * b2), $MachinePrecision], 5e-266], N[Not[LessEqual[N[(b1 * b2), $MachinePrecision], 5e+133]], $MachinePrecision]], t$95$0, N[(a1 * N[(a2 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	85.5%
Target	85.9%
Herbie	92.7%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 b1 b2) < -1.00000000000000002e198 or -4.9999999999999997e-113 < (*.f64 b1 b2) < 4.99999999999999992e-266 or 4.99999999999999961e133 < (*.f64 b1 b2)

   1. Initial program 79.4%

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

   2. Simplified 94.3%

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

      [Start] 79.4%	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 94.3%	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

   if -1.00000000000000002e198 < (*.f64 b1 b2) < -4.9999999999999997e-113

   1. Initial program 95.2%

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

   2. Simplified 83.1%

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

      [Start] 95.2%	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 83.1%	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

   3. Applied egg-rr 93.9%

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

      [Start] 83.1%	\[ \frac{a1}{b1} \cdot \frac{a2}{b2} \]
      frac-times [=>] 95.2%	\[ \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]
      *-commutative [=>] 95.2%	\[ \frac{\color{blue}{a2 \cdot a1}}{b1 \cdot b2} \]
      associate-/l* [=>] 93.9%	\[ \color{blue}{\frac{a2}{\frac{b1 \cdot b2}{a1}}} \]

   if 4.99999999999999992e-266 < (*.f64 b1 b2) < 4.99999999999999961e133

   1. Initial program 91.6%

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

   2. Simplified 83.3%

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

      [Start] 91.6%	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-/l* [=>] 91.5%	\[ \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]
      *-commutative [=>] 91.5%	\[ \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]
      associate-/l* [=>] 83.3%	\[ \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

   3. Applied egg-rr 90.1%

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

      [Start] 83.3%	\[ \frac{a1}{\frac{b2}{\frac{a2}{b1}}} \]
      clear-num [=>] 83.3%	\[ \color{blue}{\frac{1}{\frac{\frac{b2}{\frac{a2}{b1}}}{a1}}} \]
      associate-/r/ [=>] 81.9%	\[ \color{blue}{\frac{1}{\frac{b2}{\frac{a2}{b1}}} \cdot a1} \]
      clear-num [<=] 81.8%	\[ \color{blue}{\frac{\frac{a2}{b1}}{b2}} \cdot a1 \]
      associate-/l/ [=>] 90.1%	\[ \color{blue}{\frac{a2}{b2 \cdot b1}} \cdot a1 \]
      *-commutative [<=] 90.1%	\[ \frac{a2}{\color{blue}{b1 \cdot b2}} \cdot a1 \]

3. Recombined 3 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+198}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-266} \lor \neg \left(b1 \cdot b2 \leq 5 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.7%
Cost	1489

\[\begin{array}{l} t_0 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-266} \lor \neg \left(b1 \cdot b2 \leq 5 \cdot 10^{+133}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]

Alternative 2
Accuracy	92.9%
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5 \cdot 10^{+156} \lor \neg \left(b1 \cdot b2 \leq -1 \cdot 10^{-189}\right) \land \left(b1 \cdot b2 \leq 5 \cdot 10^{-266} \lor \neg \left(b1 \cdot b2 \leq 5 \cdot 10^{+133}\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]

Alternative 3
Accuracy	89.9%
Cost	1480

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-279}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 4
Accuracy	86.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.45 \cdot 10^{+165}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array} \]

Alternative 5
Accuracy	85.9%
Cost	448

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

Reproduce

herbie shell --seed 2023271 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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