Quotient of products

Average Error: 11.4 → 4.7

Time: 4.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2.428659076582579 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 1.6417374084 \cdot 10^{-314}:\\ \;\;\;\;\frac{\frac{a2 \cdot \frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;t_0 \leq 9.12141700165575 \cdot 10^{+273}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

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 -2.428659076582579e-287)
     t_0
     (if (<= t_0 1.6417374084e-314)
       (/ (/ (* a2 (/ a1 (* (cbrt b1) (cbrt b1)))) (cbrt b1)) b2)
       (if (<= t_0 9.12141700165575e+273)
         (/ 1.0 (/ (* b1 b2) (* a1 a2)))
         (* (/ a1 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 <= -2.428659076582579e-287) {
		tmp = t_0;
	} else if (t_0 <= 1.6417374084e-314) {
		tmp = ((a2 * (a1 / (cbrt(b1) * cbrt(b1)))) / cbrt(b1)) / b2;
	} else if (t_0 <= 9.12141700165575e+273) {
		tmp = 1.0 / ((b1 * b2) / (a1 * a2));
	} else {
		tmp = (a1 / 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 <= -2.428659076582579e-287) {
		tmp = t_0;
	} else if (t_0 <= 1.6417374084e-314) {
		tmp = ((a2 * (a1 / (Math.cbrt(b1) * Math.cbrt(b1)))) / Math.cbrt(b1)) / b2;
	} else if (t_0 <= 9.12141700165575e+273) {
		tmp = 1.0 / ((b1 * b2) / (a1 * a2));
	} else {
		tmp = (a1 / 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 <= -2.428659076582579e-287)
		tmp = t_0;
	elseif (t_0 <= 1.6417374084e-314)
		tmp = Float64(Float64(Float64(a2 * Float64(a1 / Float64(cbrt(b1) * cbrt(b1)))) / cbrt(b1)) / b2);
	elseif (t_0 <= 9.12141700165575e+273)
		tmp = Float64(1.0 / Float64(Float64(b1 * b2) / Float64(a1 * a2)));
	else
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	end
	return 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, -2.428659076582579e-287], t$95$0, If[LessEqual[t$95$0, 1.6417374084e-314], N[(N[(N[(a2 * N[(a1 / N[(N[Power[b1, 1/3], $MachinePrecision] * N[Power[b1, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b1, 1/3], $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[t$95$0, 9.12141700165575e+273], N[(1.0 / N[(N[(b1 * b2), $MachinePrecision] / N[(a1 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.0
Herbie	4.7

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.42865907658257895e-287

   1. Initial program 7.8

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

   if -2.42865907658257895e-287 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.6417374084e-314

   1. Initial program 13.0

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

   2. Applied associate-/r*_binary64 6.5

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

   3. Applied add-cube-cbrt_binary64 6.6

      \[\leadsto \frac{\frac{a1 \cdot a2}{\color{blue}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \sqrt[3]{b1}}}}{b2} \]

   4. Applied associate-/r*_binary64 6.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{a1 \cdot a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b1}}}}{b2} \]

   5. Simplified 4.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot a2}}{\sqrt[3]{b1}}}{b2} \]

   if 1.6417374084e-314 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 9.1214170016557509e273

   1. Initial program 0.9

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

   2. Applied clear-num_binary64 1.0

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

   if 9.1214170016557509e273 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 55.4

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

   2. Applied times-frac_binary64 8.0

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.428659076582579 \cdot 10^{-287}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.6417374084 \cdot 10^{-314}:\\ \;\;\;\;\frac{\frac{a2 \cdot \frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 9.12141700165575 \cdot 10^{+273}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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