Quotient of products

Average Error: 11.3 → 5.3

Time: 5.4s

Precision: binary64

Cost: 2008

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{if}\;a1 \cdot a2 \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-235}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 4 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (/ (* a1 a2) b2) b1)))
   (if (<= (* a1 a2) -1e+227)
     (/ a2 (* b2 (/ b1 a1)))
     (if (<= (* a1 a2) -2e-175)
       t_0
       (if (<= (* a1 a2) 1e-235)
         (/ a1 (* b2 (/ b1 a2)))
         (if (<= (* a1 a2) 4e-94)
           t_0
           (if (<= (* a1 a2) 2e+74)
             (/ (/ (* a1 a2) b1) b2)
             (if (<= (* a1 a2) 5e+242) t_0 (* (/ a2 b2) (/ a1 b1))))))))))

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) / b2) / b1;
	double tmp;
	if ((a1 * a2) <= -1e+227) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if ((a1 * a2) <= -2e-175) {
		tmp = t_0;
	} else if ((a1 * a2) <= 1e-235) {
		tmp = a1 / (b2 * (b1 / a2));
	} else if ((a1 * a2) <= 4e-94) {
		tmp = t_0;
	} else if ((a1 * a2) <= 2e+74) {
		tmp = ((a1 * a2) / b1) / b2;
	} else if ((a1 * a2) <= 5e+242) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	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) / b2) / b1
    if ((a1 * a2) <= (-1d+227)) then
        tmp = a2 / (b2 * (b1 / a1))
    else if ((a1 * a2) <= (-2d-175)) then
        tmp = t_0
    else if ((a1 * a2) <= 1d-235) then
        tmp = a1 / (b2 * (b1 / a2))
    else if ((a1 * a2) <= 4d-94) then
        tmp = t_0
    else if ((a1 * a2) <= 2d+74) then
        tmp = ((a1 * a2) / b1) / b2
    else if ((a1 * a2) <= 5d+242) then
        tmp = t_0
    else
        tmp = (a2 / b2) * (a1 / b1)
    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) / b2) / b1;
	double tmp;
	if ((a1 * a2) <= -1e+227) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if ((a1 * a2) <= -2e-175) {
		tmp = t_0;
	} else if ((a1 * a2) <= 1e-235) {
		tmp = a1 / (b2 * (b1 / a2));
	} else if ((a1 * a2) <= 4e-94) {
		tmp = t_0;
	} else if ((a1 * a2) <= 2e+74) {
		tmp = ((a1 * a2) / b1) / b2;
	} else if ((a1 * a2) <= 5e+242) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	return tmp;
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

↓

def code(a1, a2, b1, b2):
	t_0 = ((a1 * a2) / b2) / b1
	tmp = 0
	if (a1 * a2) <= -1e+227:
		tmp = a2 / (b2 * (b1 / a1))
	elif (a1 * a2) <= -2e-175:
		tmp = t_0
	elif (a1 * a2) <= 1e-235:
		tmp = a1 / (b2 * (b1 / a2))
	elif (a1 * a2) <= 4e-94:
		tmp = t_0
	elif (a1 * a2) <= 2e+74:
		tmp = ((a1 * a2) / b1) / b2
	elif (a1 * a2) <= 5e+242:
		tmp = t_0
	else:
		tmp = (a2 / b2) * (a1 / b1)
	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(Float64(a1 * a2) / b2) / b1)
	tmp = 0.0
	if (Float64(a1 * a2) <= -1e+227)
		tmp = Float64(a2 / Float64(b2 * Float64(b1 / a1)));
	elseif (Float64(a1 * a2) <= -2e-175)
		tmp = t_0;
	elseif (Float64(a1 * a2) <= 1e-235)
		tmp = Float64(a1 / Float64(b2 * Float64(b1 / a2)));
	elseif (Float64(a1 * a2) <= 4e-94)
		tmp = t_0;
	elseif (Float64(a1 * a2) <= 2e+74)
		tmp = Float64(Float64(Float64(a1 * a2) / b1) / b2);
	elseif (Float64(a1 * a2) <= 5e+242)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 / b2) * Float64(a1 / b1));
	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) / b2) / b1;
	tmp = 0.0;
	if ((a1 * a2) <= -1e+227)
		tmp = a2 / (b2 * (b1 / a1));
	elseif ((a1 * a2) <= -2e-175)
		tmp = t_0;
	elseif ((a1 * a2) <= 1e-235)
		tmp = a1 / (b2 * (b1 / a2));
	elseif ((a1 * a2) <= 4e-94)
		tmp = t_0;
	elseif ((a1 * a2) <= 2e+74)
		tmp = ((a1 * a2) / b1) / b2;
	elseif ((a1 * a2) <= 5e+242)
		tmp = t_0;
	else
		tmp = (a2 / b2) * (a1 / b1);
	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[(N[(a1 * a2), $MachinePrecision] / b2), $MachinePrecision] / b1), $MachinePrecision]}, If[LessEqual[N[(a1 * a2), $MachinePrecision], -1e+227], N[(a2 / N[(b2 * N[(b1 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a1 * a2), $MachinePrecision], -2e-175], t$95$0, If[LessEqual[N[(a1 * a2), $MachinePrecision], 1e-235], N[(a1 / N[(b2 * N[(b1 / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a1 * a2), $MachinePrecision], 4e-94], t$95$0, If[LessEqual[N[(a1 * a2), $MachinePrecision], 2e+74], N[(N[(N[(a1 * a2), $MachinePrecision] / b1), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[N[(a1 * a2), $MachinePrecision], 5e+242], t$95$0, N[(N[(a2 / b2), $MachinePrecision] * N[(a1 / b1), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	11.3
Target	11.1
Herbie	5.3

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 a1 a2) < -1.0000000000000001e227

   1. Initial program 38.2

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

   2. Applied egg-rr 9.7

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

   3. Applied egg-rr 8.6

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

   if -1.0000000000000001e227 < (*.f64 a1 a2) < -2e-175 or 9.9999999999999996e-236 < (*.f64 a1 a2) < 3.9999999999999998e-94 or 1.9999999999999999e74 < (*.f64 a1 a2) < 5.0000000000000004e242

   1. Initial program 5.3

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

   2. Simplified 11.3

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 57 points increase in error, 49 points decrease in error

   3. Applied egg-rr 5.5

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

   if -2e-175 < (*.f64 a1 a2) < 9.9999999999999996e-236

   1. Initial program 14.7

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

   2. Simplified 7.2

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 57 points increase in error, 49 points decrease in error

   3. Applied egg-rr 4.5

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

   4. Applied egg-rr 4.8

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

   if 3.9999999999999998e-94 < (*.f64 a1 a2) < 1.9999999999999999e74

   1. Initial program 2.3

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

   2. Simplified 11.4

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 57 points increase in error, 49 points decrease in error

   3. Applied egg-rr 2.0

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

   if 5.0000000000000004e242 < (*.f64 a1 a2)

   1. Initial program 41.1

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

   2. Applied egg-rr 10.7

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-235}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5
Cost	2512

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	5.6
Cost	2008

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b2}}{b1}\\ t_1 := \frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{if}\;a1 \cdot a2 \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \cdot a2 \leq 5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.3
Cost	2008

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b2}}{b1}\\ t_1 := \frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{if}\;a1 \cdot a2 \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a1 \cdot a2 \leq 4 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	5.5
Cost	1488

\[\begin{array}{l} t_0 := \frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{if}\;a1 \cdot a2 \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -1.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	11.1
Cost	976

\[\begin{array}{l} t_0 := \frac{a2}{b2 \cdot \frac{b1}{a1}}\\ t_1 := \frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{if}\;b1 \leq -7.107643639711117 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -1.5943390752560764 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	11.2
Cost	976

\[\begin{array}{l} t_0 := \frac{a1}{b2 \cdot \frac{b1}{a2}}\\ t_1 := \frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{if}\;b1 \leq -5.244479890414689 \cdot 10^{+245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq 3 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array} \]

Alternative 7
Error	11.4
Cost	976

\[\begin{array}{l} t_0 := \frac{a1}{b2 \cdot \frac{b1}{a2}}\\ t_1 := \frac{a2}{\frac{b1 \cdot b2}{a1}}\\ \mathbf{if}\;b1 \leq -9.70638525022811 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq 3 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \end{array} \]

Alternative 8
Error	11.0
Cost	448

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

Reproduce

herbie shell --seed 2022294 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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