Quotient of products

Average Error: 11.1 → 3.4

Time: 3.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a1}{b2} + \frac{a1}{b2}\right) \cdot \frac{0.5}{\frac{b1}{a2}}\\ \end{array} \]

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 (- INFINITY))
     (/ (* a1 (/ a2 b1)) b2)
     (if (<= t_0 -2e-265)
       (/ (* a1 a2) (* b2 b1))
       (if (<= t_0 0.0)
         (* a1 (/ (/ a2 b1) b2))
         (if (<= t_0 5e+240)
           (* (* a1 a2) (/ (/ 1.0 b1) b2))
           (* (+ (/ a1 b2) (/ a1 b2)) (/ 0.5 (/ b1 a2)))))))))

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 <= -((double) INFINITY)) {
		tmp = (a1 * (a2 / b1)) / b2;
	} else if (t_0 <= -2e-265) {
		tmp = (a1 * a2) / (b2 * b1);
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+240) {
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	} else {
		tmp = ((a1 / b2) + (a1 / b2)) * (0.5 / (b1 / a2));
	}
	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 <= -Double.POSITIVE_INFINITY) {
		tmp = (a1 * (a2 / b1)) / b2;
	} else if (t_0 <= -2e-265) {
		tmp = (a1 * a2) / (b2 * b1);
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (t_0 <= 5e+240) {
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	} else {
		tmp = ((a1 / b2) + (a1 / b2)) * (0.5 / (b1 / a2));
	}
	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)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (a1 * (a2 / b1)) / b2
	elif t_0 <= -2e-265:
		tmp = (a1 * a2) / (b2 * b1)
	elif t_0 <= 0.0:
		tmp = a1 * ((a2 / b1) / b2)
	elif t_0 <= 5e+240:
		tmp = (a1 * a2) * ((1.0 / b1) / b2)
	else:
		tmp = ((a1 / b2) + (a1 / b2)) * (0.5 / (b1 / a2))
	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 <= Float64(-Inf))
		tmp = Float64(Float64(a1 * Float64(a2 / b1)) / b2);
	elseif (t_0 <= -2e-265)
		tmp = Float64(Float64(a1 * a2) / Float64(b2 * b1));
	elseif (t_0 <= 0.0)
		tmp = Float64(a1 * Float64(Float64(a2 / b1) / b2));
	elseif (t_0 <= 5e+240)
		tmp = Float64(Float64(a1 * a2) * Float64(Float64(1.0 / b1) / b2));
	else
		tmp = Float64(Float64(Float64(a1 / b2) + Float64(a1 / b2)) * Float64(0.5 / Float64(b1 / a2)));
	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);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (a1 * (a2 / b1)) / b2;
	elseif (t_0 <= -2e-265)
		tmp = (a1 * a2) / (b2 * b1);
	elseif (t_0 <= 0.0)
		tmp = a1 * ((a2 / b1) / b2);
	elseif (t_0 <= 5e+240)
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	else
		tmp = ((a1 / b2) + (a1 / b2)) * (0.5 / (b1 / a2));
	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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(a1 * N[(a2 / b1), $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[t$95$0, -2e-265], N[(N[(a1 * a2), $MachinePrecision] / N[(b2 * b1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(a1 * N[(N[(a2 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+240], N[(N[(a1 * a2), $MachinePrecision] * N[(N[(1.0 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 / b2), $MachinePrecision] + N[(a1 / b2), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b1 / a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.1
Target	11.7
Herbie	3.4

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

Derivation

1. Split input into 5 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 17.0

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

   3. Applied egg-rr 17.0

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.99999999999999997e-265

   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. Taylor expanded in a1 around 0 0.8

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

   if -1.99999999999999997e-265 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 12.5

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

   2. Applied egg-rr 4.2

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

   3. Applied egg-rr 4.2

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

   4. Applied egg-rr 4.1

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.0000000000000003e240

   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. Taylor expanded in b1 around 0 1.1

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

   4. Simplified 1.2

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

   if 5.0000000000000003e240 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 48.0

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

   2. Applied egg-rr 16.6

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

   3. Applied egg-rr 16.5

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

   4. Applied egg-rr 10.4

      \[\leadsto \color{blue}{\left(\frac{a1}{b2} + \frac{a1}{b2}\right) \cdot \frac{0.5}{\frac{b1}{a2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a1}{b2} + \frac{a1}{b2}\right) \cdot \frac{0.5}{\frac{b1}{a2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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