Quotient of products

Average Error: 11.4 → 3.3

Time: 4.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;t_0 \leq -3.6097324435693325 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2.096455237501871 \cdot 10^{+303}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ b2 (/ a2 b1)))))
   (if (<= t_0 (- INFINITY))
     (* (/ a2 b2) (/ a1 b1))
     (if (<= t_0 -3.6097324435693325e-277)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 2.096455237501871e+303) 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 / (b2 / (a2 / b1));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (a2 / b2) * (a1 / b1);
	} else if (t_0 <= -3.6097324435693325e-277) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2.096455237501871e+303) {
		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 / (b2 / (a2 / b1));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (a2 / b2) * (a1 / b1);
	} else if (t_0 <= -3.6097324435693325e-277) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2.096455237501871e+303) {
		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 / (b2 / (a2 / b1))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (a2 / b2) * (a1 / b1)
	elif t_0 <= -3.6097324435693325e-277:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2.096455237501871e+303:
		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(b2 / Float64(a2 / b1)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(a2 / b2) * Float64(a1 / b1));
	elseif (t_0 <= -3.6097324435693325e-277)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2.096455237501871e+303)
		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 / (b2 / (a2 / b1));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (a2 / b2) * (a1 / b1);
	elseif (t_0 <= -3.6097324435693325e-277)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2.096455237501871e+303)
		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[(b2 / N[(a2 / b1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(a2 / b2), $MachinePrecision] * N[(a1 / b1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -3.6097324435693325e-277], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2.096455237501871e+303], t$95$0, t$95$1]]]]]]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.5
Herbie	3.3

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

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. Simplified 32.4

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

   3. Taylor expanded in a1 around 0 64.0

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

   4. Simplified 10.4

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.60973244356933245e-277 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.0964552375018711e303

   1. Initial program 0.9

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

   2. Simplified 7.7

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

   3. Applied egg-rr 8.1

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

   4. Applied egg-rr 8.1

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

   5. Applied egg-rr 8.1

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

   6. Applied egg-rr 0.9

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

   if -3.60973244356933245e-277 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 2.0964552375018711e303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 21.7

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

   2. Simplified 14.7

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

   3. Applied egg-rr 6.3

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.6097324435693325 \cdot 10^{-277}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.096455237501871 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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