Quotient of products

Average Error: 11.3 → 3.5

Time: 3.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}\\ \end{array} \]

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 (- INFINITY))
     (/ a1 (* b1 (/ b2 a2)))
     (if (<= t_0 -1e-318)
       t_0
       (if (<= t_0 2e-270)
         (/ (/ a1 (/ b1 a2)) b2)
         (if (<= t_0 2e+279) t_0 (* a1 (/ 1.0 (/ 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 <= -((double) INFINITY)) {
		tmp = a1 / (b1 * (b2 / a2));
	} else if (t_0 <= -1e-318) {
		tmp = t_0;
	} else if (t_0 <= 2e-270) {
		tmp = (a1 / (b1 / a2)) / b2;
	} else if (t_0 <= 2e+279) {
		tmp = t_0;
	} else {
		tmp = a1 * (1.0 / (b1 / (a2 / b2)));
	}
	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) {
		tmp = a1 / (b1 * (b2 / a2));
	} else if (t_0 <= -1e-318) {
		tmp = t_0;
	} else if (t_0 <= 2e-270) {
		tmp = (a1 / (b1 / a2)) / b2;
	} else if (t_0 <= 2e+279) {
		tmp = t_0;
	} else {
		tmp = a1 * (1.0 / (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 <= -math.inf:
		tmp = a1 / (b1 * (b2 / a2))
	elif t_0 <= -1e-318:
		tmp = t_0
	elif t_0 <= 2e-270:
		tmp = (a1 / (b1 / a2)) / b2
	elif t_0 <= 2e+279:
		tmp = t_0
	else:
		tmp = a1 * (1.0 / (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 <= Float64(-Inf))
		tmp = Float64(a1 / Float64(b1 * Float64(b2 / a2)));
	elseif (t_0 <= -1e-318)
		tmp = t_0;
	elseif (t_0 <= 2e-270)
		tmp = Float64(Float64(a1 / Float64(b1 / a2)) / b2);
	elseif (t_0 <= 2e+279)
		tmp = t_0;
	else
		tmp = Float64(a1 * Float64(1.0 / Float64(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 <= -Inf)
		tmp = a1 / (b1 * (b2 / a2));
	elseif (t_0 <= -1e-318)
		tmp = t_0;
	elseif (t_0 <= 2e-270)
		tmp = (a1 / (b1 / a2)) / b2;
	elseif (t_0 <= 2e+279)
		tmp = t_0;
	else
		tmp = a1 * (1.0 / (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, (-Infinity)], N[(a1 / N[(b1 * N[(b2 / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-318], t$95$0, If[LessEqual[t$95$0, 2e-270], N[(N[(a1 / N[(b1 / a2), $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[t$95$0, 2e+279], t$95$0, N[(a1 * N[(1.0 / N[(b1 / N[(a2 / b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.3
Target	11.1
Herbie	3.5

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

   1. Initial program 64.0

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

   2. Applied egg-rr 16.8

      \[\leadsto \color{blue}{a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}} \]

   3. Applied egg-rr 16.0

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.9999875e-319 or 2.0000000000000001e-270 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.00000000000000012e279

   1. Initial program 0.9

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

   if -9.9999875e-319 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.0000000000000001e-270

   1. Initial program 13.1

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

   2. Applied egg-rr 4.4

      \[\leadsto \color{blue}{a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}} \]

   3. Applied egg-rr 4.0

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

   if 2.00000000000000012e279 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 56.9

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

   2. Applied egg-rr 14.6

      \[\leadsto \color{blue}{a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022204 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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