Quotient of products

Average Error: 11.3 → 2.9

Time: 8.5s

Precision: binary64

Cost: 2640

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+284}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a2 \cdot \frac{a1}{b2}\right)\\ \end{array} \]

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 b2) (/ a2 b1))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -1e-316)
       t_0
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 1e+284) t_0 (* (/ 1.0 b1) (* a2 (/ a1 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 t_1 = (a1 / b2) * (a2 / b1);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -1e-316) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+284) {
		tmp = t_0;
	} else {
		tmp = (1.0 / b1) * (a2 * (a1 / 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 t_1 = (a1 / b2) * (a2 / b1);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -1e-316) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+284) {
		tmp = t_0;
	} else {
		tmp = (1.0 / b1) * (a2 * (a1 / 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)
	t_1 = (a1 / b2) * (a2 / b1)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= -1e-316:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+284:
		tmp = t_0
	else:
		tmp = (1.0 / b1) * (a2 * (a1 / 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))
	t_1 = Float64(Float64(a1 / b2) * Float64(a2 / b1))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -1e-316)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+284)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / b1) * Float64(a2 * Float64(a1 / 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);
	t_1 = (a1 / b2) * (a2 / b1);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= -1e-316)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+284)
		tmp = t_0;
	else
		tmp = (1.0 / b1) * (a2 * (a1 / 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]}, Block[{t$95$1 = N[(N[(a1 / b2), $MachinePrecision] * N[(a2 / b1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -1e-316], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+284], t$95$0, N[(N[(1.0 / b1), $MachinePrecision] * N[(a2 * N[(a1 / b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	11.3
Target	11.1
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or -9.999999837e-317 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 17.9

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

   2. Simplified 3.1

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

      [Start] 17.9	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      rational.json-simplify-46 [=>] 8.4	\[ \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}} \]
      rational.json-simplify-44 [=>] 8.5	\[ \color{blue}{\frac{\frac{a1 \cdot a2}{b2}}{b1}} \]
      rational.json-simplify-49 [=>] 4.9	\[ \frac{\color{blue}{a2 \cdot \frac{a1}{b2}}}{b1} \]
      rational.json-simplify-49 [=>] 3.1	\[ \color{blue}{\frac{a1}{b2} \cdot \frac{a2}{b1}} \]

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.999999837e-317 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.00000000000000008e284

   1. Initial program 0.9

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

   if 1.00000000000000008e284 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 57.7

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

   2. Simplified 43.6

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

      [Start] 57.7	\[ \frac{a1 \cdot a2}{b1 \cdot b2} \]
      rational.json-simplify-2 [=>] 57.7	\[ \frac{\color{blue}{a2 \cdot a1}}{b1 \cdot b2} \]
      rational.json-simplify-49 [=>] 43.6	\[ \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]

   3. Applied egg-rr 15.5

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{b1}}{\frac{\frac{b2}{a2}}{a1} + \frac{\frac{b2}{a2}}{a1}}} \]

   4. Simplified 16.0

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

      [Start] 15.5	\[ \frac{2 \cdot \frac{1}{b1}}{\frac{\frac{b2}{a2}}{a1} + \frac{\frac{b2}{a2}}{a1}} \]
      rational.json-simplify-49 [=>] 16.3	\[ \color{blue}{\frac{1}{b1} \cdot \frac{2}{\frac{\frac{b2}{a2}}{a1} + \frac{\frac{b2}{a2}}{a1}}} \]
      metadata-eval [<=] 16.3	\[ \frac{1}{b1} \cdot \frac{\color{blue}{1 + 1}}{\frac{\frac{b2}{a2}}{a1} + \frac{\frac{b2}{a2}}{a1}} \]
      rational.json-simplify-35 [<=] 16.2	\[ \frac{1}{b1} \cdot \color{blue}{\frac{1}{\frac{\frac{b2}{a2}}{a1}}} \]
      rational.json-simplify-47 [=>] 47.0	\[ \frac{1}{b1} \cdot \frac{1}{\color{blue}{\frac{b2}{a2 \cdot a1}}} \]
      rational.json-simplify-2 [<=] 47.0	\[ \frac{1}{b1} \cdot \frac{1}{\frac{b2}{\color{blue}{a1 \cdot a2}}} \]
      rational.json-simplify-61 [=>] 47.0	\[ \frac{1}{b1} \cdot \color{blue}{\frac{a1 \cdot a2}{\frac{b2}{1}}} \]
      rational.json-simplify-7 [=>] 47.0	\[ \frac{1}{b1} \cdot \frac{a1 \cdot a2}{\color{blue}{b2}} \]
      rational.json-simplify-49 [=>] 16.0	\[ \frac{1}{b1} \cdot \color{blue}{\left(a2 \cdot \frac{a1}{b2}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+284}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a2 \cdot \frac{a1}{b2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.6
Cost	2512

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternative 2
Error	5.8
Cost	1488

\[\begin{array}{l} t_0 := a1 \cdot \frac{a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.7
Cost	1488

\[\begin{array}{l} t_0 := a1 \cdot \frac{a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternative 4
Error	6.5
Cost	1488

\[\begin{array}{l} t_0 := \frac{a2}{\frac{b1 \cdot b2}{a1}}\\ t_1 := \frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{-173}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	10.1
Cost	1372

\[\begin{array}{l} t_0 := \frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{if}\;b2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \leq -6.2 \cdot 10^{-225}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b2 \leq 1.25 \cdot 10^{-168}:\\ \;\;\;\;\frac{a2}{\frac{b1}{a1} \cdot b2}\\ \mathbf{elif}\;b2 \leq 5 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \leq 1.7 \cdot 10^{+65}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;b2 \leq 3.8 \cdot 10^{+229}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{elif}\;b2 \leq 1.12 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \]

Alternative 6
Error	11.3
Cost	448

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

Reproduce

herbie shell --seed 2023064 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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