Quotient of products

Average Accuracy: 82.5% → 94.9%

Time: 6.5s

Precision: binary64

Cost: 2512

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{a2}{\frac{b2}{a1}}}{b1}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

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+293)
     (* (/ a2 b1) (/ a1 b2))
     (if (<= t_0 -5e-308)
       t_0
       (if (<= t_0 0.0)
         (/ (/ a2 (/ b2 a1)) b1)
         (if (<= t_0 5e+258) t_0 (/ a1 (* b1 (/ b2 a2)))))))))

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+293) {
		tmp = (a2 / b1) * (a1 / b2);
	} else if (t_0 <= -5e-308) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 / (b2 / a1)) / b1;
	} else if (t_0 <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = a1 / (b1 * (b2 / a2));
	}
	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+293)) then
        tmp = (a2 / b1) * (a1 / b2)
    else if (t_0 <= (-5d-308)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a2 / (b2 / a1)) / b1
    else if (t_0 <= 5d+258) then
        tmp = t_0
    else
        tmp = a1 / (b1 * (b2 / a2))
    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+293) {
		tmp = (a2 / b1) * (a1 / b2);
	} else if (t_0 <= -5e-308) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 / (b2 / a1)) / b1;
	} else if (t_0 <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = a1 / (b1 * (b2 / a2));
	}
	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+293:
		tmp = (a2 / b1) * (a1 / b2)
	elif t_0 <= -5e-308:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a2 / (b2 / a1)) / b1
	elif t_0 <= 5e+258:
		tmp = t_0
	else:
		tmp = a1 / (b1 * (b2 / a2))
	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+293)
		tmp = Float64(Float64(a2 / b1) * Float64(a1 / b2));
	elseif (t_0 <= -5e-308)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a2 / Float64(b2 / a1)) / b1);
	elseif (t_0 <= 5e+258)
		tmp = t_0;
	else
		tmp = Float64(a1 / Float64(b1 * Float64(b2 / a2)));
	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+293)
		tmp = (a2 / b1) * (a1 / b2);
	elseif (t_0 <= -5e-308)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a2 / (b2 / a1)) / b1;
	elseif (t_0 <= 5e+258)
		tmp = t_0;
	else
		tmp = a1 / (b1 * (b2 / a2));
	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+293], N[(N[(a2 / b1), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-308], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a2 / N[(b2 / a1), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision], If[LessEqual[t$95$0, 5e+258], t$95$0, N[(a1 / N[(b1 * N[(b2 / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	82.5%
Target	81.5%
Herbie	94.9%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9999999999999998e293

   1. Initial program 9.5%

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

   2. Simplified 75.3%

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

      [Start] 9.5	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-/l* [=>] 54.7	\[ \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]
      *-commutative [=>] 54.7	\[ \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]
      associate-/l* [=>] 75.3	\[ \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

   3. Taylor expanded in a1 around 0 9.5%

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

   4. Simplified 80.0%

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

      [Start] 9.5	\[ \frac{a1 \cdot a2}{b2 \cdot b1} \]
      times-frac [=>] 80.0	\[ \color{blue}{\frac{a1}{b2} \cdot \frac{a2}{b1}} \]
      *-commutative [=>] 80.0	\[ \color{blue}{\frac{a2}{b1} \cdot \frac{a1}{b2}} \]

   if -1.9999999999999998e293 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.99999999999999955e-308 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5e258

   1. Initial program 98.7%

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

   if -4.99999999999999955e-308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 78.9%

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

   2. Simplified 96.2%

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

      [Start] 78.9	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 96.2	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

   3. Applied egg-rr 94.6%

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

      [Start] 96.2	\[ \frac{a1}{b1} \cdot \frac{a2}{b2} \]
      associate-*l/ [=>] 94.6	\[ \color{blue}{\frac{a1 \cdot \frac{a2}{b2}}{b1}} \]

   4. Taylor expanded in a1 around 0 89.5%

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

   5. Simplified 94.3%

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

      [Start] 89.5	\[ \frac{\frac{a1 \cdot a2}{b2}}{b1} \]
      *-commutative [=>] 89.5	\[ \frac{\frac{\color{blue}{a2 \cdot a1}}{b2}}{b1} \]
      associate-/l* [=>] 94.3	\[ \frac{\color{blue}{\frac{a2}{\frac{b2}{a1}}}}{b1} \]

   if 5e258 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 20.8%

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

   2. Simplified 84.1%

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

      [Start] 20.8	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 84.1	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

   3. Applied egg-rr 78.7%

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

      [Start] 84.1	\[ \frac{a1}{b1} \cdot \frac{a2}{b2} \]
      *-commutative [=>] 84.1	\[ \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
      clear-num [=>] 83.9	\[ \color{blue}{\frac{1}{\frac{b2}{a2}}} \cdot \frac{a1}{b1} \]
      frac-times [=>] 78.7	\[ \color{blue}{\frac{1 \cdot a1}{\frac{b2}{a2} \cdot b1}} \]
      *-un-lft-identity [<=] 78.7	\[ \frac{\color{blue}{a1}}{\frac{b2}{a2} \cdot b1} \]

3. Recombined 4 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a2}{\frac{b2}{a1}}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+258}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.7%
Cost	2513

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-289} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 5 \cdot 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

Alternative 2
Accuracy	94.8%
Cost	2512

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

Alternative 3
Accuracy	82.4%
Cost	845

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.6 \cdot 10^{+281} \lor \neg \left(a1 \leq -7.5 \cdot 10^{+124}\right) \land a1 \leq -3.3 \cdot 10^{-273}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]

Alternative 4
Accuracy	82.5%
Cost	844

\[\begin{array}{l} t_0 := \frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{if}\;a1 \leq -6.5 \cdot 10^{+206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \leq -1.55 \cdot 10^{+103}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;a1 \leq -2.5 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

Alternative 5
Accuracy	81.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;a1 \leq -8.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;a1 \leq -9 \cdot 10^{-273}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array} \]

Alternative 6
Accuracy	82.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array} \]

Alternative 7
Accuracy	82.8%
Cost	448

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

Reproduce

herbie shell --seed 2023151 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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