Quotient of products

Average Error: 10.6 → 3.0

Time: 3.7s

Precision: binary64

Cost: 2512

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+271}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 b1) (/ b2 a1))
     (if (<= t_0 -2e-308)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 5e+271) 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 / b1) / (b2 / a1);
	} else if (t_0 <= -2e-308) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+271) {
		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 / b1) / (b2 / a1);
	} else if (t_0 <= -2e-308) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+271) {
		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 / b1) / (b2 / a1)
	elif t_0 <= -2e-308:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+271:
		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(a1 / Float64(b1 / Float64(a2 / b2)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(a2 / b1) / Float64(b2 / a1));
	elseif (t_0 <= -2e-308)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+271)
		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 / b1) / (b2 / a1);
	elseif (t_0 <= -2e-308)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+271)
		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[(a1 / N[(b1 / N[(a2 / b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(a2 / b1), $MachinePrecision] / N[(b2 / a1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-308], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+271], t$95$0, t$95$1]]]]]]

Target

Original	10.6
Target	11.0
Herbie	3.0

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

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 8.9

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

   3. Applied egg-rr 8.6

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9999999999999998e-308 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.0000000000000003e271

   1. Initial program 0.7

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

   if -1.9999999999999998e-308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 5.0000000000000003e271 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 21.2

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

   2. Simplified 13.9

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 44 points increase in error, 41 points decrease in error

   3. Applied egg-rr 5.9

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	1240

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b2}}{b1}\\ t_1 := \frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ t_2 := \frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{if}\;a2 \leq -6.481964030080444 \cdot 10^{-260}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;a2 \leq 9.679537110650904 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a2 \leq 2.478716341095882 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 10^{+185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a2 \leq 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a2 \leq 5.5 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	11.5
Cost	1240

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b2}}{b1}\\ t_1 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ t_2 := a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{if}\;b1 \leq -6.56227236502032 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;b1 \leq -27657140846.473663:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \leq -3.399460119634546 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b1 \leq -1 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \leq 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	11.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;a1 \leq -4.21308739775774 \cdot 10^{-87}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \end{array} \]

Alternative 4
Error	11.0
Cost	448

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

Alternative 5
Error	11.1
Cost	448

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

Reproduce

herbie shell --seed 2022301 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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