Quotient of products

Average Error: 11.7 → 3.0

Time: 4.7s

Precision: binary64

Cost: 2512

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+289}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq 10^{+272}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -2e+289)
     (* (/ a2 b2) (/ a1 b1))
     (if (<= t_0 -5e-321)
       t_0
       (if (<= t_0 0.0)
         (/ (/ a1 b1) (/ b2 a2))
         (if (<= t_0 1e+272) t_0 (/ (* a2 (/ a1 b2)) b1)))))))

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 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+289) {
		tmp = (a2 / b2) * (a1 / b1);
	} else if (t_0 <= -5e-321) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) / (b2 / a2);
	} else if (t_0 <= 1e+272) {
		tmp = t_0;
	} else {
		tmp = (a2 * (a1 / b2)) / b1;
	}
	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 * a2) / (b1 * b2)
    if (t_0 <= (-2d+289)) then
        tmp = (a2 / b2) * (a1 / b1)
    else if (t_0 <= (-5d-321)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a1 / b1) / (b2 / a2)
    else if (t_0 <= 1d+272) then
        tmp = t_0
    else
        tmp = (a2 * (a1 / b2)) / b1
    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 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+289) {
		tmp = (a2 / b2) * (a1 / b1);
	} else if (t_0 <= -5e-321) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) / (b2 / a2);
	} else if (t_0 <= 1e+272) {
		tmp = t_0;
	} else {
		tmp = (a2 * (a1 / b2)) / b1;
	}
	return tmp;
}

Python

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

↓

def code(a1, a2, b1, b2):
	t_0 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_0 <= -2e+289:
		tmp = (a2 / b2) * (a1 / b1)
	elif t_0 <= -5e-321:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a1 / b1) / (b2 / a2)
	elif t_0 <= 1e+272:
		tmp = t_0
	else:
		tmp = (a2 * (a1 / b2)) / b1
	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 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -2e+289)
		tmp = Float64(Float64(a2 / b2) * Float64(a1 / b1));
	elseif (t_0 <= -5e-321)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a1 / b1) / Float64(b2 / a2));
	elseif (t_0 <= 1e+272)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 * Float64(a1 / b2)) / b1);
	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 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -2e+289)
		tmp = (a2 / b2) * (a1 / b1);
	elseif (t_0 <= -5e-321)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a1 / b1) / (b2 / a2);
	elseif (t_0 <= 1e+272)
		tmp = t_0;
	else
		tmp = (a2 * (a1 / b2)) / b1;
	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 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+289], N[(N[(a2 / b2), $MachinePrecision] * N[(a1 / b1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-321], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a1 / b1), $MachinePrecision] / N[(b2 / a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+272], t$95$0, N[(N[(a2 * N[(a1 / b2), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision]]]]]]

Target

Original	11.7
Target	11.4
Herbie	3.0

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.0000000000000001e289

   1. Initial program 55.6

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

   2. Applied egg-rr 13.3

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

   if -2.0000000000000001e289 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.99994e-321 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.0000000000000001e272

   1. Initial program 0.9

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

   if -4.99994e-321 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.9

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

   2. Applied egg-rr 2.5

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

   3. Applied egg-rr 2.6

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

   if 1.0000000000000001e272 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 57.1

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

   2. Simplified 45.1

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 44 points increase in error, 41 points decrease in error

   3. Applied egg-rr 46.2

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

   4. Applied egg-rr 13.8

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+289}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+272}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.6
Cost	1488

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-296}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array} \]

Alternative 2
Error	11.7
Cost	1240

\[\begin{array}{l} t_0 := \frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ t_1 := \frac{a2 \cdot \frac{a1}{b2}}{b1}\\ t_2 := \frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{if}\;b1 \leq -7.695126657006692 \cdot 10^{+199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -4.2437960029836874 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \leq -9.393784703302856 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b1 \leq -5 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.0
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{if}\;a1 \cdot a2 \leq -5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 5 \cdot 10^{-189}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	11.4
Cost	844

\[\begin{array}{l} t_0 := \frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ t_1 := \frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{if}\;b2 \leq 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b2 \leq 3 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \leq 7.381579423550374 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	11.4
Cost	448

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

Reproduce

herbie shell --seed 2022313 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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