Quotient of products

Average Error: 11.3 → 2.8

Time: 4.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 b2) (/ b1 a2))))
   (if (<= t_0 (- INFINITY))
     (/ a2 (* b2 (/ b1 a1)))
     (if (<= t_0 -2e-296)
       t_0
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 5e+289) (* (* a1 a2) (/ 1.0 (* b1 b2))) 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 / b2) / (b1 / a2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if (t_0 <= -2e-296) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+289) {
		tmp = (a1 * a2) * (1.0 / (b1 * b2));
	} 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 / b2) / (b1 / a2);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = a2 / (b2 * (b1 / a1));
	} else if (t_0 <= -2e-296) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+289) {
		tmp = (a1 * a2) * (1.0 / (b1 * b2));
	} 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 / b2) / (b1 / a2)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = a2 / (b2 * (b1 / a1))
	elif t_0 <= -2e-296:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+289:
		tmp = (a1 * a2) * (1.0 / (b1 * b2))
	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 / b2) / Float64(b1 / a2))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(a2 / Float64(b2 * Float64(b1 / a1)));
	elseif (t_0 <= -2e-296)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+289)
		tmp = Float64(Float64(a1 * a2) * Float64(1.0 / Float64(b1 * b2)));
	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 / b2) / (b1 / a2);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = a2 / (b2 * (b1 / a1));
	elseif (t_0 <= -2e-296)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+289)
		tmp = (a1 * a2) * (1.0 / (b1 * b2));
	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 / b2), $MachinePrecision] / N[(b1 / a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(a2 / N[(b2 * N[(b1 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-296], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+289], N[(N[(a1 * a2), $MachinePrecision] * N[(1.0 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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	11.6
Herbie	2.8

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

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 10.4

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

   3. Applied egg-rr 17.1

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

   4. Taylor expanded in b1 around 0 31.5

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

   5. Simplified 18.3

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2e-296

   1. Initial program 0.8

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

   2. Applied egg-rr 15.6

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

   3. Applied egg-rr 0.8

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

   if -2e-296 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 5.00000000000000031e289 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 21.1

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

   2. Applied egg-rr 3.7

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

   3. Applied egg-rr 3.8

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.00000000000000031e289

   1. Initial program 0.8

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

   2. Applied egg-rr 1.1

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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