Quotient of products

Average Error: 11.9 → 3.2

Time: 4.7s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-314}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

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 -5e+294)
     (/ a2 (* b2 (/ b1 a1)))
     (if (<= t_0 -1e-309)
       t_0
       (if (<= t_0 5e-314)
         (* a1 (/ (/ a2 b1) b2))
         (if (<= t_0 5e+265) t_0 (* (/ a1 b1) (/ a2 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 <= -5e+294) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if (t_0 <= -1e-309) {
		tmp = t_0;
	} else if (t_0 <= 5e-314) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+265) {
		tmp = t_0;
	} else {
		tmp = (a1 / b1) * (a2 / 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 <= (-5d+294)) then
        tmp = a2 / (b2 * (b1 / a1))
    else if (t_0 <= (-1d-309)) then
        tmp = t_0
    else if (t_0 <= 5d-314) then
        tmp = a1 * ((a2 / b1) / b2)
    else if (t_0 <= 5d+265) then
        tmp = t_0
    else
        tmp = (a1 / b1) * (a2 / 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 <= -5e+294) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if (t_0 <= -1e-309) {
		tmp = t_0;
	} else if (t_0 <= 5e-314) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+265) {
		tmp = t_0;
	} else {
		tmp = (a1 / b1) * (a2 / 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 <= -5e+294:
		tmp = a2 / (b2 * (b1 / a1))
	elif t_0 <= -1e-309:
		tmp = t_0
	elif t_0 <= 5e-314:
		tmp = a1 * ((a2 / b1) / b2)
	elif t_0 <= 5e+265:
		tmp = t_0
	else:
		tmp = (a1 / b1) * (a2 / 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 <= -5e+294)
		tmp = Float64(a2 / Float64(b2 * Float64(b1 / a1)));
	elseif (t_0 <= -1e-309)
		tmp = t_0;
	elseif (t_0 <= 5e-314)
		tmp = Float64(a1 * Float64(Float64(a2 / b1) / b2));
	elseif (t_0 <= 5e+265)
		tmp = t_0;
	else
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / 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 <= -5e+294)
		tmp = a2 / (b2 * (b1 / a1));
	elseif (t_0 <= -1e-309)
		tmp = t_0;
	elseif (t_0 <= 5e-314)
		tmp = a1 * ((a2 / b1) / b2);
	elseif (t_0 <= 5e+265)
		tmp = t_0;
	else
		tmp = (a1 / b1) * (a2 / 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, -5e+294], N[(a2 / N[(b2 * N[(b1 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-309], t$95$0, If[LessEqual[t$95$0, 5e-314], N[(a1 * N[(N[(a2 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+265], t$95$0, N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.9
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)) < -4.9999999999999999e294

   1. Initial program 56.6

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

   2. Simplified 28.7

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

      [Start] 56.6	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-*l/ [<=] 28.7	\[ \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]
      *-commutative [=>] 28.7	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]

   3. Applied egg-rr 14.9

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

   if -4.9999999999999999e294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.000000000000002e-309 or 4.99999999982e-314 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.0000000000000002e265

   1. Initial program 0.9

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

   if -1.000000000000002e-309 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.99999999982e-314

   1. Initial program 15.0

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

   2. Simplified 3.7

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

      [Start] 15.0	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-/r* [=>] 6.9	\[ \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}} \]
      *-commutative [=>] 6.9	\[ \frac{\frac{\color{blue}{a2 \cdot a1}}{b1}}{b2} \]
      associate-/l* [=>] 3.7	\[ \frac{\color{blue}{\frac{a2}{\frac{b1}{a1}}}}{b2} \]

   3. Applied egg-rr 4.0

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

   if 5.0000000000000002e265 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 52.8

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

   2. Simplified 9.6

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

      [Start] 52.8	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 9.6	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{+294}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{-314}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.7
Cost	1243

\[\begin{array}{l} \mathbf{if}\;b2 \leq -2.2 \cdot 10^{+257} \lor \neg \left(b2 \leq -3.2 \cdot 10^{+140}\right) \land \left(b2 \leq -4.2 \cdot 10^{-288} \lor \neg \left(b2 \leq 5.5 \cdot 10^{-287}\right) \land \left(b2 \leq 5.1 \cdot 10^{-189} \lor \neg \left(b2 \leq 3 \cdot 10^{+27}\right)\right)\right):\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 2
Error	7.5
Cost	1228

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 4 \cdot 10^{-227}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array} \]

Alternative 3
Error	11.9
Cost	448

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

Reproduce

herbie shell --seed 2023125 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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