Quotient of products

Average Error: 10.9 → 2.6

Time: 3.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b2 \cdot b1}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+284}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (/ a2 b1) (/ b2 a1))))
   (if (<= t_0 (- INFINITY))
     (/ (* a2 (/ a1 b2)) b1)
     (if (<= t_0 -1e-310)
       (* (* a1 a2) (/ 1.0 (* b2 b1)))
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 1e+284) (/ (* a1 a2) (* b2 b1)) 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 = (a2 / b1) / (b2 / a1);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= -1e-310) {
		tmp = (a1 * a2) * (1.0 / (b2 * b1));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+284) {
		tmp = (a1 * a2) / (b2 * b1);
	} 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 = (a2 / b1) / (b2 / a1);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= -1e-310) {
		tmp = (a1 * a2) * (1.0 / (b2 * b1));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+284) {
		tmp = (a1 * a2) / (b2 * b1);
	} 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 = (a2 / b1) / (b2 / a1)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (a2 * (a1 / b2)) / b1
	elif t_0 <= -1e-310:
		tmp = (a1 * a2) * (1.0 / (b2 * b1))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+284:
		tmp = (a1 * a2) / (b2 * b1)
	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(a2 / b1) / Float64(b2 / a1))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(a2 * Float64(a1 / b2)) / b1);
	elseif (t_0 <= -1e-310)
		tmp = Float64(Float64(a1 * a2) * Float64(1.0 / Float64(b2 * b1)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+284)
		tmp = Float64(Float64(a1 * a2) / Float64(b2 * b1));
	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 = (a2 / b1) / (b2 / a1);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (a2 * (a1 / b2)) / b1;
	elseif (t_0 <= -1e-310)
		tmp = (a1 * a2) * (1.0 / (b2 * b1));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+284)
		tmp = (a1 * a2) / (b2 * b1);
	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[(a2 / b1), $MachinePrecision] / N[(b2 / a1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(a2 * N[(a1 / b2), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision], If[LessEqual[t$95$0, -1e-310], N[(N[(a1 * a2), $MachinePrecision] * N[(1.0 / N[(b2 * b1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+284], N[(N[(a1 * a2), $MachinePrecision] / N[(b2 * b1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.9
Target	11.2
Herbie	2.6

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

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. Taylor expanded in a1 around 0 64.0

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

   3. Simplified 20.5

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

   4. Applied egg-rr 30.4

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

   5. Applied egg-rr 17.4

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.999999999999969e-311

   1. Initial program 0.8

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

   2. Taylor expanded in a1 around 0 0.8

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

   3. Simplified 15.0

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

   4. Applied egg-rr 7.7

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

   5. Applied egg-rr 1.3

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

   if -9.999999999999969e-311 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 1.00000000000000008e284 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 21.3

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

   2. Simplified 5.1

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

   3. Applied egg-rr 3.6

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

   if 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.00000000000000008e284

   1. Initial program 0.8

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

   2. Taylor expanded in a1 around 0 0.8

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

   3. Simplified 14.7

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

   4. Applied egg-rr 8.2

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

   5. Taylor expanded in a1 around 0 0.8

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b2 \cdot b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+284}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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