Quotient of products

Average Error: 11.3 → 2.1

Time: 4.3s

Precision: binary64

Cost: 2514

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq -5 \cdot 10^{-316} \lor \neg \left(t_0 \leq 2 \cdot 10^{-306}\right) \land t_0 \leq 6 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (or (<= t_0 (- INFINITY))
           (not
            (or (<= t_0 -5e-316) (and (not (<= t_0 2e-306)) (<= t_0 6e+299)))))
     (/ (/ a1 b1) (/ b2 a2))
     t_0)))

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 <= -((double) INFINITY)) || !((t_0 <= -5e-316) || (!(t_0 <= 2e-306) && (t_0 <= 6e+299)))) {
		tmp = (a1 / b1) / (b2 / a2);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) || !((t_0 <= -5e-316) || (!(t_0 <= 2e-306) && (t_0 <= 6e+299)))) {
		tmp = (a1 / b1) / (b2 / a2);
	} else {
		tmp = t_0;
	}
	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 <= -math.inf) or not ((t_0 <= -5e-316) or (not (t_0 <= 2e-306) and (t_0 <= 6e+299))):
		tmp = (a1 / b1) / (b2 / a2)
	else:
		tmp = t_0
	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 <= Float64(-Inf)) || !((t_0 <= -5e-316) || (!(t_0 <= 2e-306) && (t_0 <= 6e+299))))
		tmp = Float64(Float64(a1 / b1) / Float64(b2 / a2));
	else
		tmp = t_0;
	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 <= -Inf) || ~(((t_0 <= -5e-316) || (~((t_0 <= 2e-306)) && (t_0 <= 6e+299)))))
		tmp = (a1 / b1) / (b2 / a2);
	else
		tmp = t_0;
	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[Or[LessEqual[t$95$0, (-Infinity)], N[Not[Or[LessEqual[t$95$0, -5e-316], And[N[Not[LessEqual[t$95$0, 2e-306]], $MachinePrecision], LessEqual[t$95$0, 6e+299]]]], $MachinePrecision]], N[(N[(a1 / b1), $MachinePrecision] / N[(b2 / a2), $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	11.3
Target	11.1
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or -5.000000017e-316 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.00000000000000006e-306 or 6.0000000000000003e299 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 25.3

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

   2. Simplified 15.5

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

      [Start] 25.3	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      associate-*l/ [<=] 15.5	\[ \color{blue}{\frac{a1}{b1 \cdot b2} \cdot a2} \]
      *-commutative [=>] 15.5	\[ \color{blue}{a2 \cdot \frac{a1}{b1 \cdot b2}} \]

   3. Applied egg-rr 3.9

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.000000017e-316 or 2.00000000000000006e-306 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 6.0000000000000003e299

   1. Initial program 0.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-316} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2 \cdot 10^{-306}\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 6 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	2514

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq -5 \cdot 10^{-316} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 6 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.7
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+287} \lor \neg \left(b1 \cdot b2 \leq -2 \cdot 10^{-60} \lor \neg \left(b1 \cdot b2 \leq 2 \cdot 10^{-239}\right) \land b1 \cdot b2 \leq 3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 3
Error	6.1
Cost	1489

\[\begin{array}{l} t_0 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5 \cdot 10^{-269} \lor \neg \left(b1 \cdot b2 \leq 2 \cdot 10^{+112}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \end{array} \]

Alternative 4
Error	11.2
Cost	448

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

Reproduce

herbie shell --seed 2023012 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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