Quotient of products

Average Error: 11.2 → 5.5

Time: 5.1s

Precision: binary64

Cost: 1489

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{a2}{b1} \cdot a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{-292} \lor \neg \left(b1 \cdot b2 \leq 2 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) -1e+131)
   (/ (* (/ a2 b1) a1) b2)
   (if (<= (* b1 b2) -2e-250)
     (/ a2 (/ (* b1 b2) a1))
     (if (or (<= (* b1 b2) 1e-292) (not (<= (* b1 b2) 2e+146)))
       (/ (/ a2 (/ b1 a1)) b2)
       (/ (* a2 a1) (* 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) <= -1e+131) {
		tmp = ((a2 / b1) * a1) / b2;
	} else if ((b1 * b2) <= -2e-250) {
		tmp = a2 / ((b1 * b2) / a1);
	} else if (((b1 * b2) <= 1e-292) || !((b1 * b2) <= 2e+146)) {
		tmp = (a2 / (b1 / a1)) / b2;
	} else {
		tmp = (a2 * a1) / (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) <= (-1d+131)) then
        tmp = ((a2 / b1) * a1) / b2
    else if ((b1 * b2) <= (-2d-250)) then
        tmp = a2 / ((b1 * b2) / a1)
    else if (((b1 * b2) <= 1d-292) .or. (.not. ((b1 * b2) <= 2d+146))) then
        tmp = (a2 / (b1 / a1)) / b2
    else
        tmp = (a2 * a1) / (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) <= -1e+131) {
		tmp = ((a2 / b1) * a1) / b2;
	} else if ((b1 * b2) <= -2e-250) {
		tmp = a2 / ((b1 * b2) / a1);
	} else if (((b1 * b2) <= 1e-292) || !((b1 * b2) <= 2e+146)) {
		tmp = (a2 / (b1 / a1)) / b2;
	} else {
		tmp = (a2 * a1) / (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) <= -1e+131:
		tmp = ((a2 / b1) * a1) / b2
	elif (b1 * b2) <= -2e-250:
		tmp = a2 / ((b1 * b2) / a1)
	elif ((b1 * b2) <= 1e-292) or not ((b1 * b2) <= 2e+146):
		tmp = (a2 / (b1 / a1)) / b2
	else:
		tmp = (a2 * a1) / (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) <= -1e+131)
		tmp = Float64(Float64(Float64(a2 / b1) * a1) / b2);
	elseif (Float64(b1 * b2) <= -2e-250)
		tmp = Float64(a2 / Float64(Float64(b1 * b2) / a1));
	elseif ((Float64(b1 * b2) <= 1e-292) || !(Float64(b1 * b2) <= 2e+146))
		tmp = Float64(Float64(a2 / Float64(b1 / a1)) / b2);
	else
		tmp = Float64(Float64(a2 * a1) / Float64(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) <= -1e+131)
		tmp = ((a2 / b1) * a1) / b2;
	elseif ((b1 * b2) <= -2e-250)
		tmp = a2 / ((b1 * b2) / a1);
	elseif (((b1 * b2) <= 1e-292) || ~(((b1 * b2) <= 2e+146)))
		tmp = (a2 / (b1 / a1)) / b2;
	else
		tmp = (a2 * a1) / (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], -1e+131], N[(N[(N[(a2 / b1), $MachinePrecision] * a1), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], -2e-250], N[(a2 / N[(N[(b1 * b2), $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(b1 * b2), $MachinePrecision], 1e-292], N[Not[LessEqual[N[(b1 * b2), $MachinePrecision], 2e+146]], $MachinePrecision]], N[(N[(a2 / N[(b1 / a1), $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision], N[(N[(a2 * a1), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	11.2
Target	11.9
Herbie	5.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 b1 b2) < -9.9999999999999991e130

   1. Initial program 13.4

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

   2. Simplified 13.0

      \[\leadsto \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]
      Proof
      (*.f64 a2 (/.f64 a1 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a2 a1) (*.f64 b1 b2))): 0 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a1 a2)) (*.f64 b1 b2)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 6.4

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

   if -9.9999999999999991e130 < (*.f64 b1 b2) < -2.0000000000000001e-250

   1. Initial program 4.0

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

   2. Simplified 15.8

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
      Proof
      (*.f64 a2 (/.f64 a1 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a2 a1) (*.f64 b1 b2))): 0 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a1 a2)) (*.f64 b1 b2)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 4.3

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

   if -2.0000000000000001e-250 < (*.f64 b1 b2) < 1.0000000000000001e-292 or 1.99999999999999987e146 < (*.f64 b1 b2)

   1. Initial program 23.6

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

   2. Simplified 6.5

      \[\leadsto \color{blue}{\frac{\frac{a2}{\frac{b1}{a1}}}{b2}} \]
      Proof
      (*.f64 a2 (/.f64 a1 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a2 a1) (*.f64 b1 b2))): 0 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a1 a2)) (*.f64 b1 b2)): 0 points increase in error, 0 points decrease in error

   if 1.0000000000000001e-292 < (*.f64 b1 b2) < 1.99999999999999987e146

   1. Initial program 5.0

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{a2}{b1} \cdot a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{-292} \lor \neg \left(b1 \cdot b2 \leq 2 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.9
Cost	1997

\[\begin{array}{l} t_0 := \frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-287} \lor \neg \left(t_0 \leq 5 \cdot 10^{-301}\right) \land t_0 \leq 10^{+308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array} \]

Alternative 2
Error	5.7
Cost	1748

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b1 \cdot b2}\\ t_1 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 3
Error	5.6
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+179} \lor \neg \left(b1 \cdot b2 \leq -2 \cdot 10^{-250} \lor \neg \left(b1 \cdot b2 \leq 5 \cdot 10^{-316}\right) \land b1 \cdot b2 \leq 10^{+122}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 4
Error	5.2
Cost	1489

\[\begin{array}{l} t_0 := \frac{\frac{a2}{b1} \cdot a1}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-316} \lor \neg \left(b1 \cdot b2 \leq 4 \cdot 10^{+256}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 5
Error	5.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{a2}{\frac{b1 \cdot b2}{a1}}\\ t_1 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 6
Error	11.2
Cost	448

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

Reproduce

herbie shell --seed 2022343 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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