Quotient of products

Average Error: 11.3 → 3.1

Time: 3.8s

Precision: binary64

\[[a1, a2] = \mathsf{sort}([a1, a2]) \[b1, b2] = \mathsf{sort}([b1, b2]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{if}\;t_0 \leq -3.490612777261886 \cdot 10^{+283}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1.166 \cdot 10^{-321}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq 3.7245197859932976 \cdot 10^{+273}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (/ a2 b1) b2))))
   (if (<= t_0 -3.490612777261886e+283)
     t_1
     (if (<= t_0 -1.166e-321)
       t_0
       (if (<= t_0 0.0)
         (* (/ a1 b1) (/ a2 b2))
         (if (<= t_0 3.7245197859932976e+273) 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 * ((a2 / b1) / b2);
	double tmp;
	if (t_0 <= -3.490612777261886e+283) {
		tmp = t_1;
	} else if (t_0 <= -1.166e-321) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 3.7245197859932976e+273) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 * a2) / (b1 * b2)
    t_1 = a1 * ((a2 / b1) / b2)
    if (t_0 <= (-3.490612777261886d+283)) then
        tmp = t_1
    else if (t_0 <= (-1.166d-321)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (a1 / b1) * (a2 / b2)
    else if (t_0 <= 3.7245197859932976d+273) then
        tmp = t_0
    else
        tmp = t_1
    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 * a2) / (b1 * b2);
	double t_1 = a1 * ((a2 / b1) / b2);
	double tmp;
	if (t_0 <= -3.490612777261886e+283) {
		tmp = t_1;
	} else if (t_0 <= -1.166e-321) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 3.7245197859932976e+273) {
		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 * ((a2 / b1) / b2)
	tmp = 0
	if t_0 <= -3.490612777261886e+283:
		tmp = t_1
	elif t_0 <= -1.166e-321:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (a1 / b1) * (a2 / b2)
	elif t_0 <= 3.7245197859932976e+273:
		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(Float64(a2 / b1) / b2))
	tmp = 0.0
	if (t_0 <= -3.490612777261886e+283)
		tmp = t_1;
	elseif (t_0 <= -1.166e-321)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(a1 / b1) * Float64(a2 / b2));
	elseif (t_0 <= 3.7245197859932976e+273)
		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 * ((a2 / b1) / b2);
	tmp = 0.0;
	if (t_0 <= -3.490612777261886e+283)
		tmp = t_1;
	elseif (t_0 <= -1.166e-321)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (a1 / b1) * (a2 / b2);
	elseif (t_0 <= 3.7245197859932976e+273)
		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[(N[(a2 / b1), $MachinePrecision] / b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3.490612777261886e+283], t$95$1, If[LessEqual[t$95$0, -1.166e-321], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(a1 / b1), $MachinePrecision] * N[(a2 / b2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 3.7245197859932976e+273], 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	11.3
Target	11.1
Herbie	3.1

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.49061277726188584e283 or 3.72451978599329762e273 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 56.3

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

   2. Applied div-inv_binary64 56.3

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

   3. Applied associate-*l*_binary64 39.7

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

   4. Applied *-un-lft-identity_binary64 39.7

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

   5. Applied times-frac_binary64 39.1

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

   6. Applied associate-*r*_binary64 16.3

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

   7. Simplified 16.2

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

   8. Applied un-div-inv_binary64 16.2

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

   if -3.49061277726188584e283 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.16599e-321 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 3.72451978599329762e273

   1. Initial program 0.8

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

   if -1.16599e-321 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.1

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

   2. Applied times-frac_binary64 2.2

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.490612777261886 \cdot 10^{+283}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.166 \cdot 10^{-321}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.7245197859932976 \cdot 10^{+273}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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