Quotient of products

Average Error: 11.3 → 3.4

Time: 4.5s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b2}{a1}}}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -2e+301)
     (/ (* a2 (/ a1 b2)) b1)
     (if (<= t_0 -5e-285)
       t_0
       (if (<= t_0 0.0)
         (* (/ a1 b1) (/ a2 b2))
         (if (<= t_0 5e+265) 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) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+301) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= -5e-285) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 5e+265) {
		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) / (b1 * b2)
    if (t_0 <= (-2d+301)) then
        tmp = (a2 * (a1 / b2)) / b1
    else if (t_0 <= (-5d-285)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a1 / b1) * (a2 / b2)
    else if (t_0 <= 5d+265) 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) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+301) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= -5e-285) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 5e+265) {
		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) / (b1 * b2)
	tmp = 0
	if t_0 <= -2e+301:
		tmp = (a2 * (a1 / b2)) / b1
	elif t_0 <= -5e-285:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a1 / b1) * (a2 / b2)
	elif t_0 <= 5e+265:
		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(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -2e+301)
		tmp = Float64(Float64(a2 * Float64(a1 / b2)) / b1);
	elseif (t_0 <= -5e-285)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	elseif (t_0 <= 5e+265)
		tmp = t_0;
	else
		tmp = Float64(Float64(a2 / Float64(b2 / 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) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -2e+301)
		tmp = (a2 * (a1 / b2)) / b1;
	elseif (t_0 <= -5e-285)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a1 / b1) * (a2 / b2);
	elseif (t_0 <= 5e+265)
		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[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+301], N[(N[(a2 * N[(a1 / b2), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision], If[LessEqual[t$95$0, -5e-285], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+265], t$95$0, N[(N[(a2 / N[(b2 / a1), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision]]]]]]

Target

Original	11.3
Target	11.4
Herbie	3.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.00000000000000011e301

   1. Initial program 59.7

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

   2. Simplified 30.2

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

      [Start] 59.7	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-*l/ [<=] 30.2	\[ \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]
      *-commutative [=>] 30.2	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]

   3. Applied egg-rr 18.7

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

   if -2.00000000000000011e301 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.00000000000000018e-285 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.0000000000000002e265

   1. Initial program 0.8

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

   if -5.00000000000000018e-285 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 12.7

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

   2. Simplified 3.1

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

      [Start] 12.7	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      times-frac [=>] 3.1	\[ \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

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

   1. Initial program 53.9

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

   2. Simplified 42.6

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

      [Start] 53.9	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-*l/ [<=] 42.6	\[ \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]
      *-commutative [=>] 42.6	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]

   3. Applied egg-rr 15.5

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b2}{a1}}}{b1}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	2513

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-285} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 2 \cdot 10^{+274}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 2
Error	3.4
Cost	2512

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.4
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+290} \lor \neg \left(b1 \cdot b2 \leq -1 \cdot 10^{-275} \lor \neg \left(b1 \cdot b2 \leq 0\right) \land b1 \cdot b2 \leq 10^{+308}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 4
Error	11.4
Cost	448

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

Reproduce

herbie shell --seed 2023034 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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