Quotient of products

Average Error: 11.3 → 6.2

Time: 3.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-266}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-195}:\\ \;\;\;\;{\left(\frac{b1}{\frac{a2}{\frac{b2}{a1}}}\right)}^{-1}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) -5e+221)
   (/ a2 (* b2 (/ b1 a1)))
   (if (<= (* b1 b2) -5e-266)
     (* (* a2 a1) (/ 1.0 (* b1 b2)))
     (if (<= (* b1 b2) 5e-195)
       (pow (/ b1 (/ a2 (/ b2 a1))) -1.0)
       (if (<= (* b1 b2) 2e+112)
         (* a2 (/ a1 (* b1 b2)))
         (* (/ a1 b2) (/ a2 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 tmp;
	if ((b1 * b2) <= -5e+221) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if ((b1 * b2) <= -5e-266) {
		tmp = (a2 * a1) * (1.0 / (b1 * b2));
	} else if ((b1 * b2) <= 5e-195) {
		tmp = pow((b1 / (a2 / (b2 / a1))), -1.0);
	} else if ((b1 * b2) <= 2e+112) {
		tmp = a2 * (a1 / (b1 * b2));
	} else {
		tmp = (a1 / b2) * (a2 / 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) :: tmp
    if ((b1 * b2) <= (-5d+221)) then
        tmp = a2 / (b2 * (b1 / a1))
    else if ((b1 * b2) <= (-5d-266)) then
        tmp = (a2 * a1) * (1.0d0 / (b1 * b2))
    else if ((b1 * b2) <= 5d-195) then
        tmp = (b1 / (a2 / (b2 / a1))) ** (-1.0d0)
    else if ((b1 * b2) <= 2d+112) then
        tmp = a2 * (a1 / (b1 * b2))
    else
        tmp = (a1 / b2) * (a2 / 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 tmp;
	if ((b1 * b2) <= -5e+221) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if ((b1 * b2) <= -5e-266) {
		tmp = (a2 * a1) * (1.0 / (b1 * b2));
	} else if ((b1 * b2) <= 5e-195) {
		tmp = Math.pow((b1 / (a2 / (b2 / a1))), -1.0);
	} else if ((b1 * b2) <= 2e+112) {
		tmp = a2 * (a1 / (b1 * b2));
	} else {
		tmp = (a1 / b2) * (a2 / b1);
	}
	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) <= -5e+221:
		tmp = a2 / (b2 * (b1 / a1))
	elif (b1 * b2) <= -5e-266:
		tmp = (a2 * a1) * (1.0 / (b1 * b2))
	elif (b1 * b2) <= 5e-195:
		tmp = math.pow((b1 / (a2 / (b2 / a1))), -1.0)
	elif (b1 * b2) <= 2e+112:
		tmp = a2 * (a1 / (b1 * b2))
	else:
		tmp = (a1 / b2) * (a2 / 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)
	tmp = 0.0
	if (Float64(b1 * b2) <= -5e+221)
		tmp = Float64(a2 / Float64(b2 * Float64(b1 / a1)));
	elseif (Float64(b1 * b2) <= -5e-266)
		tmp = Float64(Float64(a2 * a1) * Float64(1.0 / Float64(b1 * b2)));
	elseif (Float64(b1 * b2) <= 5e-195)
		tmp = Float64(b1 / Float64(a2 / Float64(b2 / a1))) ^ -1.0;
	elseif (Float64(b1 * b2) <= 2e+112)
		tmp = Float64(a2 * Float64(a1 / Float64(b1 * b2)));
	else
		tmp = Float64(Float64(a1 / b2) * Float64(a2 / 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)
	tmp = 0.0;
	if ((b1 * b2) <= -5e+221)
		tmp = a2 / (b2 * (b1 / a1));
	elseif ((b1 * b2) <= -5e-266)
		tmp = (a2 * a1) * (1.0 / (b1 * b2));
	elseif ((b1 * b2) <= 5e-195)
		tmp = (b1 / (a2 / (b2 / a1))) ^ -1.0;
	elseif ((b1 * b2) <= 2e+112)
		tmp = a2 * (a1 / (b1 * b2));
	else
		tmp = (a1 / b2) * (a2 / 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_] := If[LessEqual[N[(b1 * b2), $MachinePrecision], -5e+221], N[(a2 / N[(b2 * N[(b1 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], -5e-266], N[(N[(a2 * a1), $MachinePrecision] * N[(1.0 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], 5e-195], N[Power[N[(b1 / N[(a2 / N[(b2 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], 2e+112], N[(a2 * N[(a1 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1 / b2), $MachinePrecision] * N[(a2 / b1), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	11.3
Target	11.6
Herbie	6.2

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 b1 b2) < -5.0000000000000002e221

   1. Initial program 16.1

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

   2. Applied egg-rr 16.1

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

   3. Applied egg-rr 16.3

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

   4. Taylor expanded in b1 around 0 16.1

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

   5. Simplified 8.2

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

   if -5.0000000000000002e221 < (*.f64 b1 b2) < -4.99999999999999992e-266

   1. Initial program 5.1

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

   2. Applied egg-rr 5.1

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

   if -4.99999999999999992e-266 < (*.f64 b1 b2) < 5.00000000000000009e-195

   1. Initial program 36.4

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

   2. Applied egg-rr 38.6

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

   3. Applied egg-rr 36.4

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

   4. Applied egg-rr 9.2

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

   if 5.00000000000000009e-195 < (*.f64 b1 b2) < 1.9999999999999999e112

   1. Initial program 4.1

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

   2. Applied egg-rr 4.2

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

   if 1.9999999999999999e112 < (*.f64 b1 b2)

   1. Initial program 12.8

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

   2. Applied egg-rr 7.1

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

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-266}:\\ \;\;\;\;\left(a2 \cdot a1\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-195}:\\ \;\;\;\;{\left(\frac{b1}{\frac{a2}{\frac{b2}{a1}}}\right)}^{-1}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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