Quotient of products

Average Error: 11.2 → 3.8

Time: 4.5s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;t_0 \leq -4 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \mathbf{elif}\;t_0 \leq 10^{+272}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

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+287)
     (* a2 (/ a1 (* b1 b2)))
     (if (<= t_0 -4e-288)
       t_0
       (if (<= t_0 0.0)
         (/ (/ a2 (/ b1 a1)) b2)
         (if (<= t_0 1e+272) t_0 (/ (/ a2 b2) (/ b1 a1))))))))

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+287) {
		tmp = a2 * (a1 / (b1 * b2));
	} else if (t_0 <= -4e-288) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 / (b1 / a1)) / b2;
	} else if (t_0 <= 1e+272) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) / (b1 / a1);
	}
	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+287)) then
        tmp = a2 * (a1 / (b1 * b2))
    else if (t_0 <= (-4d-288)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a2 / (b1 / a1)) / b2
    else if (t_0 <= 1d+272) then
        tmp = t_0
    else
        tmp = (a2 / b2) / (b1 / a1)
    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+287) {
		tmp = a2 * (a1 / (b1 * b2));
	} else if (t_0 <= -4e-288) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 / (b1 / a1)) / b2;
	} else if (t_0 <= 1e+272) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) / (b1 / a1);
	}
	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+287:
		tmp = a2 * (a1 / (b1 * b2))
	elif t_0 <= -4e-288:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a2 / (b1 / a1)) / b2
	elif t_0 <= 1e+272:
		tmp = t_0
	else:
		tmp = (a2 / b2) / (b1 / a1)
	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+287)
		tmp = Float64(a2 * Float64(a1 / Float64(b1 * b2)));
	elseif (t_0 <= -4e-288)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a2 / Float64(b1 / a1)) / b2);
	elseif (t_0 <= 1e+272)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 / b2) / Float64(b1 / a1));
	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+287)
		tmp = a2 * (a1 / (b1 * b2));
	elseif (t_0 <= -4e-288)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a2 / (b1 / a1)) / b2;
	elseif (t_0 <= 1e+272)
		tmp = t_0;
	else
		tmp = (a2 / b2) / (b1 / a1);
	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+287], N[(a2 * N[(a1 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -4e-288], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a2 / N[(b1 / a1), $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[t$95$0, 1e+272], t$95$0, N[(N[(a2 / b2), $MachinePrecision] / N[(b1 / a1), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.2
Target	11.7
Herbie	3.8

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.0000000000000001e287

   1. Initial program 53.8

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

   2. Simplified 29.4

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

      [Start] 53.8	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-*l/ [<=] 29.4	\[ \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]
      *-commutative [=>] 29.4	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]

   if -1.0000000000000001e287 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.00000000000000023e-288 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.0000000000000001e272

   1. Initial program 0.8

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

   if -4.00000000000000023e-288 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 12.5

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

   2. Simplified 4.2

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

      [Start] 12.5	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-/r* [=>] 6.2	\[ \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}} \]
      *-commutative [=>] 6.2	\[ \frac{\frac{\color{blue}{a2 \cdot a1}}{b1}}{b2} \]
      associate-/l* [=>] 4.2	\[ \frac{\color{blue}{\frac{a2}{\frac{b1}{a1}}}}{b2} \]

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

   1. Initial program 55.4

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

   2. Simplified 9.9

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

      [Start] 55.4	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 9.9	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

   3. Applied egg-rr 9.5

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+272}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.2
Cost	2513

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-307} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternative 2
Error	3.5
Cost	2512

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternative 3
Error	3.5
Cost	2512

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 10^{+272}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

Alternative 4
Error	5.8
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+301} \lor \neg \left(b1 \cdot b2 \leq -1 \cdot 10^{-88} \lor \neg \left(b1 \cdot b2 \leq 10^{-244}\right) \land b1 \cdot b2 \leq 5 \cdot 10^{+278}\right):\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 5
Error	11.0
Cost	448

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

Reproduce

herbie shell --seed 2023060 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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