Quotient of products

Average Error: 10.8 → 3.2

Time: 4.8s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+261}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;t_0 \leq -4 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq 10^{+295}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -1e+261)
     (/ a2 (* b1 (/ b2 a1)))
     (if (<= t_0 -4e-258)
       t_0
       (if (<= t_0 0.0)
         (* a2 (/ (/ a1 b1) b2))
         (if (<= t_0 1e+295) t_0 (* (/ a2 b1) (/ a1 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 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -1e+261) {
		tmp = a2 / (b1 * (b2 / a1));
	} else if (t_0 <= -4e-258) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a2 * ((a1 / b1) / b2);
	} else if (t_0 <= 1e+295) {
		tmp = t_0;
	} else {
		tmp = (a2 / b1) * (a1 / 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 * a2) / (b1 * b2)
    if (t_0 <= (-1d+261)) then
        tmp = a2 / (b1 * (b2 / a1))
    else if (t_0 <= (-4d-258)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = a2 * ((a1 / b1) / b2)
    else if (t_0 <= 1d+295) then
        tmp = t_0
    else
        tmp = (a2 / b1) * (a1 / 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 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -1e+261) {
		tmp = a2 / (b1 * (b2 / a1));
	} else if (t_0 <= -4e-258) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a2 * ((a1 / b1) / b2);
	} else if (t_0 <= 1e+295) {
		tmp = t_0;
	} else {
		tmp = (a2 / b1) * (a1 / b2);
	}
	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 <= -1e+261:
		tmp = a2 / (b1 * (b2 / a1))
	elif t_0 <= -4e-258:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = a2 * ((a1 / b1) / b2)
	elif t_0 <= 1e+295:
		tmp = t_0
	else:
		tmp = (a2 / b1) * (a1 / 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 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -1e+261)
		tmp = Float64(a2 / Float64(b1 * Float64(b2 / a1)));
	elseif (t_0 <= -4e-258)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(a2 * Float64(Float64(a1 / b1) / b2));
	elseif (t_0 <= 1e+295)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 / b1) * Float64(a1 / 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 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -1e+261)
		tmp = a2 / (b1 * (b2 / a1));
	elseif (t_0 <= -4e-258)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = a2 * ((a1 / b1) / b2);
	elseif (t_0 <= 1e+295)
		tmp = t_0;
	else
		tmp = (a2 / b1) * (a1 / 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 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+261], N[(a2 / N[(b1 * N[(b2 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -4e-258], t$95$0, If[LessEqual[t$95$0, 0.0], N[(a2 * N[(N[(a1 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+295], t$95$0, N[(N[(a2 / b1), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	10.8
Target	11.2
Herbie	3.2

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.9999999999999993e260

   1. Initial program 44.4

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

   2. Simplified 24.1

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

      [Start] 44.4	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-*l/ [<=] 24.1	\[ \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]
      *-commutative [=>] 24.1	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]

   3. Applied egg-rr 18.2

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

   if -9.9999999999999993e260 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.99999999999999982e-258 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 9.9999999999999998e294

   1. Initial program 0.9

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

   if -3.99999999999999982e-258 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 11.5

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

   2. Taylor expanded in a1 around 0 11.5

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

   3. Simplified 4.0

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

      [Start] 11.5	\[ \frac{a1 \cdot a2}{b2 \cdot b1} \]
      *-commutative [=>] 11.5	\[ \frac{\color{blue}{a2 \cdot a1}}{b2 \cdot b1} \]
      *-commutative [=>] 11.5	\[ \frac{a2 \cdot a1}{\color{blue}{b1 \cdot b2}} \]
      associate-*r/ [<=] 6.7	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]
      associate-/r* [=>] 4.0	\[ a2 \cdot \color{blue}{\frac{\frac{a1}{b1}}{b2}} \]

   if 9.9999999999999998e294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 60.5

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

   2. Applied egg-rr 7.2

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{+261}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4 \cdot 10^{-258}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+295}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5 \cdot 10^{+142} \lor \neg \left(b1 \cdot b2 \leq -5 \cdot 10^{-130}\right) \land \left(b1 \cdot b2 \leq 4 \cdot 10^{-193} \lor \neg \left(b1 \cdot b2 \leq 5 \cdot 10^{+220}\right)\right):\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 2
Error	5.9
Cost	1488

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 4 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \end{array} \]

Alternative 3
Error	5.9
Cost	1488

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 4 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+233}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \end{array} \]

Alternative 4
Error	10.5
Cost	448

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

Reproduce

herbie shell --seed 2023076 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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