Quotient of products

Average Error: 11.3 → 2.8

Time: 3.8s

Precision: binary64

\[[a1, a2] = \mathsf{sort}([a1, a2]) \[b1, b2] = \mathsf{sort}([b1, b2]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -3.971449504747777 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 4.3318940274796574 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.134688516477658 \cdot 10^{+283}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))) (t_1 (* (/ a1 b1) (/ a2 b2))))
   (if (<= t_0 (- INFINITY))
     (/ (* a2 (/ a1 b2)) b1)
     (if (<= t_0 -3.971449504747777e-256)
       t_0
       (if (<= t_0 4.3318940274796574e-291)
         t_1
         (if (<= t_0 1.134688516477658e+283) t_0 t_1))))))

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 t_1 = (a1 / b1) * (a2 / b2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= -3.971449504747777e-256) {
		tmp = t_0;
	} else if (t_0 <= 4.3318940274796574e-291) {
		tmp = t_1;
	} else if (t_0 <= 1.134688516477658e+283) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 t_1 = (a1 / b1) * (a2 / b2);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= -3.971449504747777e-256) {
		tmp = t_0;
	} else if (t_0 <= 4.3318940274796574e-291) {
		tmp = t_1;
	} else if (t_0 <= 1.134688516477658e+283) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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)
	t_1 = (a1 / b1) * (a2 / b2)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (a2 * (a1 / b2)) / b1
	elif t_0 <= -3.971449504747777e-256:
		tmp = t_0
	elif t_0 <= 4.3318940274796574e-291:
		tmp = t_1
	elif t_0 <= 1.134688516477658e+283:
		tmp = t_0
	else:
		tmp = t_1
	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))
	t_1 = Float64(Float64(a1 / b1) * Float64(a2 / b2))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(a2 * Float64(a1 / b2)) / b1);
	elseif (t_0 <= -3.971449504747777e-256)
		tmp = t_0;
	elseif (t_0 <= 4.3318940274796574e-291)
		tmp = t_1;
	elseif (t_0 <= 1.134688516477658e+283)
		tmp = t_0;
	else
		tmp = t_1;
	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);
	t_1 = (a1 / b1) * (a2 / b2);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (a2 * (a1 / b2)) / b1;
	elseif (t_0 <= -3.971449504747777e-256)
		tmp = t_0;
	elseif (t_0 <= 4.3318940274796574e-291)
		tmp = t_1;
	elseif (t_0 <= 1.134688516477658e+283)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(a2 * N[(a1 / b2), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision], If[LessEqual[t$95$0, -3.971449504747777e-256], t$95$0, If[LessEqual[t$95$0, 4.3318940274796574e-291], t$95$1, If[LessEqual[t$95$0, 1.134688516477658e+283], t$95$0, t$95$1]]]]]]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.4
Herbie	2.8

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

Derivation

1. Split input into 3 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 31.3

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

   3. Applied egg-rr 16.2

      \[\leadsto \color{blue}{\left(\frac{1}{b1} \cdot \frac{a1}{b2}\right)} \cdot a2 \]

   4. Applied egg-rr 16.4

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.9714495047477771e-256 or 4.33189402747965744e-291 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.1346885164776581e283

   1. Initial program 0.8

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

   2. Applied egg-rr 7.3

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

   3. Applied egg-rr 14.8

      \[\leadsto \color{blue}{\left(\frac{1}{b1} \cdot \frac{a1}{b2}\right)} \cdot a2 \]

   4. Applied egg-rr 0.8

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

   if -3.9714495047477771e-256 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.33189402747965744e-291 or 1.1346885164776581e283 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 20.6

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

   2. Applied egg-rr 4.2

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.971449504747777 \cdot 10^{-256}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.3318940274796574 \cdot 10^{-291}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.134688516477658 \cdot 10^{+283}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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