Quotient of products

Average Error: 10.9 → 4.9

Time: 3.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+273}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1 \cdot 10^{-245}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-195}:\\ \;\;\;\;{\left(\frac{b1}{a1} \cdot \frac{b2}{a2}\right)}^{-1}\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+224}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) -2e+273)
   (/ (* a1 (/ a2 b2)) b1)
   (if (<= (* b1 b2) -1e-245)
     (/ a2 (/ (* b1 b2) a1))
     (if (<= (* b1 b2) 2e-195)
       (pow (* (/ b1 a1) (/ b2 a2)) -1.0)
       (if (<= (* b1 b2) 1e+224)
         (* a2 (/ a1 (* b1 b2)))
         (/ (* 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 tmp;
	if ((b1 * b2) <= -2e+273) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if ((b1 * b2) <= -1e-245) {
		tmp = a2 / ((b1 * b2) / a1);
	} else if ((b1 * b2) <= 2e-195) {
		tmp = pow(((b1 / a1) * (b2 / a2)), -1.0);
	} else if ((b1 * b2) <= 1e+224) {
		tmp = a2 * (a1 / (b1 * b2));
	} 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) :: tmp
    if ((b1 * b2) <= (-2d+273)) then
        tmp = (a1 * (a2 / b2)) / b1
    else if ((b1 * b2) <= (-1d-245)) then
        tmp = a2 / ((b1 * b2) / a1)
    else if ((b1 * b2) <= 2d-195) then
        tmp = ((b1 / a1) * (b2 / a2)) ** (-1.0d0)
    else if ((b1 * b2) <= 1d+224) then
        tmp = a2 * (a1 / (b1 * b2))
    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 tmp;
	if ((b1 * b2) <= -2e+273) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if ((b1 * b2) <= -1e-245) {
		tmp = a2 / ((b1 * b2) / a1);
	} else if ((b1 * b2) <= 2e-195) {
		tmp = Math.pow(((b1 / a1) * (b2 / a2)), -1.0);
	} else if ((b1 * b2) <= 1e+224) {
		tmp = a2 * (a1 / (b1 * b2));
	} 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):
	tmp = 0
	if (b1 * b2) <= -2e+273:
		tmp = (a1 * (a2 / b2)) / b1
	elif (b1 * b2) <= -1e-245:
		tmp = a2 / ((b1 * b2) / a1)
	elif (b1 * b2) <= 2e-195:
		tmp = math.pow(((b1 / a1) * (b2 / a2)), -1.0)
	elif (b1 * b2) <= 1e+224:
		tmp = a2 * (a1 / (b1 * b2))
	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)
	tmp = 0.0
	if (Float64(b1 * b2) <= -2e+273)
		tmp = Float64(Float64(a1 * Float64(a2 / b2)) / b1);
	elseif (Float64(b1 * b2) <= -1e-245)
		tmp = Float64(a2 / Float64(Float64(b1 * b2) / a1));
	elseif (Float64(b1 * b2) <= 2e-195)
		tmp = Float64(Float64(b1 / a1) * Float64(b2 / a2)) ^ -1.0;
	elseif (Float64(b1 * b2) <= 1e+224)
		tmp = Float64(a2 * Float64(a1 / Float64(b1 * b2)));
	else
		tmp = Float64(Float64(a1 * Float64(a2 / 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)
	tmp = 0.0;
	if ((b1 * b2) <= -2e+273)
		tmp = (a1 * (a2 / b2)) / b1;
	elseif ((b1 * b2) <= -1e-245)
		tmp = a2 / ((b1 * b2) / a1);
	elseif ((b1 * b2) <= 2e-195)
		tmp = ((b1 / a1) * (b2 / a2)) ^ -1.0;
	elseif ((b1 * b2) <= 1e+224)
		tmp = a2 * (a1 / (b1 * b2));
	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_] := If[LessEqual[N[(b1 * b2), $MachinePrecision], -2e+273], N[(N[(a1 * N[(a2 / b2), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], -1e-245], N[(a2 / N[(N[(b1 * b2), $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], 2e-195], N[Power[N[(N[(b1 / a1), $MachinePrecision] * N[(b2 / a2), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], 1e+224], N[(a2 * N[(a1 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1 * N[(a2 / b1), $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision]]]]]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.9
Target	11.3
Herbie	4.9

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 b1 b2) < -1.99999999999999989e273

   1. Initial program 20.2

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

   2. Applied egg-rr 2.9

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

   3. Applied egg-rr 3.1

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

   4. Taylor expanded in a2 around 0 7.7

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

   5. Simplified 3.7

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

   if -1.99999999999999989e273 < (*.f64 b1 b2) < -9.9999999999999993e-246

   1. Initial program 5.1

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

   2. Applied egg-rr 14.2

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

   3. Applied egg-rr 11.1

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

   4. Taylor expanded in b1 around 0 4.9

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

   if -9.9999999999999993e-246 < (*.f64 b1 b2) < 2.0000000000000002e-195

   1. Initial program 32.3

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

   2. Applied egg-rr 8.6

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

   if 2.0000000000000002e-195 < (*.f64 b1 b2) < 9.9999999999999997e223

   1. Initial program 4.2

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

   2. Applied egg-rr 4.2

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

   if 9.9999999999999997e223 < (*.f64 b1 b2)

   1. Initial program 16.8

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

   2. Applied egg-rr 4.3

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

   3. Applied egg-rr 4.4

      \[\leadsto \color{blue}{\frac{\frac{a2}{b1} \cdot a1}{b2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+273}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1 \cdot 10^{-245}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-195}:\\ \;\;\;\;{\left(\frac{b1}{a1} \cdot \frac{b2}{a2}\right)}^{-1}\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+224}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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