Quotient of products

Average Error: 11.3 → 4.4

Time: 3.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ t_1 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+255}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (/ a1 b2) (/ b1 a2))) (t_1 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_1 -2e-293)
     t_1
     (if (<= t_1 0.0) t_0 (if (<= t_1 2e+255) t_1 t_0)))))

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 / b2) / (b1 / a2);
	double t_1 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_1 <= -2e-293) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+255) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    code = (a1 * a2) / (b1 * b2)
end function

↓

real(8) function code(a1, a2, b1, b2)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: b1
    real(8), intent (in) :: b2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a1 / b2) / (b1 / a2)
    t_1 = (a1 * a2) / (b1 * b2)
    if (t_1 <= (-2d-293)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 2d+255) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

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 / b2) / (b1 / a2);
	double t_1 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_1 <= -2e-293) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+255) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a1, a2, b1, b2):
	return (a1 * a2) / (b1 * b2)

↓

def code(a1, a2, b1, b2):
	t_0 = (a1 / b2) / (b1 / a2)
	t_1 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_1 <= -2e-293:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+255:
		tmp = t_1
	else:
		tmp = t_0
	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 / b2) / Float64(b1 / a2))
	t_1 = Float64(Float64(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_1 <= -2e-293)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+255)
		tmp = t_1;
	else
		tmp = t_0;
	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 / b2) / (b1 / a2);
	t_1 = (a1 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_1 <= -2e-293)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+255)
		tmp = t_1;
	else
		tmp = t_0;
	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 / b2), $MachinePrecision] / N[(b1 / a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-293], t$95$1, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+255], t$95$1, t$95$0]]]]]

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.2
Herbie	4.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.0000000000000001e-293 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.99999999999999998e255

   1. Initial program 4.4

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

   if -2.0000000000000001e-293 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 1.99999999999999998e255 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 21.4

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

   2. Simplified 15.0

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

   3. Applied egg-rr 14.4

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

   4. Applied egg-rr 4.3

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

   5. Applied egg-rr 4.3

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{\frac{b1}{a2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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