Quotient of products

Average Error: 11.4 → 3.1

Time: 4.2s

Precision: binary64

Cost: 2512

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+305}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq 10^{+294}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

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 -2e+305)
     (* a2 (/ (/ a1 b2) b1))
     (if (<= t_0 -5e-270)
       t_0
       (if (<= t_0 0.0)
         (/ a2 (* b2 (/ b1 a1)))
         (if (<= t_0 1e+294) 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 <= -2e+305) {
		tmp = a2 * ((a1 / b2) / b1);
	} else if (t_0 <= -5e-270) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if (t_0 <= 1e+294) {
		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 <= (-2d+305)) then
        tmp = a2 * ((a1 / b2) / b1)
    else if (t_0 <= (-5d-270)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = a2 / (b2 * (b1 / a1))
    else if (t_0 <= 1d+294) 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 <= -2e+305) {
		tmp = a2 * ((a1 / b2) / b1);
	} else if (t_0 <= -5e-270) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if (t_0 <= 1e+294) {
		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 <= -2e+305:
		tmp = a2 * ((a1 / b2) / b1)
	elif t_0 <= -5e-270:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = a2 / (b2 * (b1 / a1))
	elif t_0 <= 1e+294:
		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 <= -2e+305)
		tmp = Float64(a2 * Float64(Float64(a1 / b2) / b1));
	elseif (t_0 <= -5e-270)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(a2 / Float64(b2 * Float64(b1 / a1)));
	elseif (t_0 <= 1e+294)
		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 <= -2e+305)
		tmp = a2 * ((a1 / b2) / b1);
	elseif (t_0 <= -5e-270)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = a2 / (b2 * (b1 / a1));
	elseif (t_0 <= 1e+294)
		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, -2e+305], N[(a2 * N[(N[(a1 / b2), $MachinePrecision] / b1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-270], t$95$0, If[LessEqual[t$95$0, 0.0], N[(a2 / N[(b2 * N[(b1 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+294], t$95$0, N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.4
Target	11.1
Herbie	3.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9999999999999999e305

   1. Initial program 62.8

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

   2. Simplified 34.1

      \[\leadsto \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]
      Proof
      (*.f64 a2 (/.f64 a1 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 a1 (*.f64 b1 b2)) a2)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 40 points increase in error, 36 points decrease in error

   3. Taylor expanded in a1 around 0 34.1

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

   4. Simplified 17.5

      \[\leadsto a2 \cdot \color{blue}{\frac{\frac{a1}{b2}}{b1}} \]
      Proof
      (/.f64 (/.f64 a1 b2) b1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 a1 (*.f64 b2 b1))): 49 points increase in error, 44 points decrease in error

   if -1.9999999999999999e305 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.9999999999999998e-270 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.00000000000000007e294

   1. Initial program 0.9

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

   if -4.9999999999999998e-270 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.6

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

   2. Simplified 3.5

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
      Proof
      (*.f64 (/.f64 a1 b1) (/.f64 a2 b2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 65 points increase in error, 55 points decrease in error

   3. Applied egg-rr 4.7

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

   if 1.00000000000000007e294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 60.3

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

   2. Simplified 6.4

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
      Proof
      (*.f64 (/.f64 a1 b1) (/.f64 a2 b2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 65 points increase in error, 55 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+305}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	1488

\[\begin{array}{l} t_0 := a1 \cdot \frac{a2}{b1 \cdot b2}\\ t_1 := a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \mathbf{if}\;b1 \cdot b2 \leq -2 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-97}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b2}}{b1}\\ \end{array} \]

Alternative 3
Error	6.8
Cost	1488

\[\begin{array}{l} t_0 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-97}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \end{array} \]

Alternative 4
Error	11.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.35 \cdot 10^{+200}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 5
Error	11.4
Cost	448

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

Reproduce

herbie shell --seed 2022339 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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