Quotient of products

Average Error: 11.3 → 2.1

Time: 3.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{if}\;t_0 \leq -1.040164306635014 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -7.4 \cdot 10^{-323}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 6.474204089623792 \cdot 10^{+306}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (/ 1.0 (/ b1 a2)) (/ a1 b2))))
   (if (<= t_0 -1.040164306635014e+306)
     t_1
     (if (<= t_0 -7.4e-323)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 6.474204089623792e+306) 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 = (1.0 / (b1 / a2)) * (a1 / b2);
	double tmp;
	if (t_0 <= -1.040164306635014e+306) {
		tmp = t_1;
	} else if (t_0 <= -7.4e-323) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 6.474204089623792e+306) {
		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 = (1.0d0 / (b1 / a2)) * (a1 / b2)
    if (t_0 <= (-1.040164306635014d+306)) then
        tmp = t_1
    else if (t_0 <= (-7.4d-323)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 6.474204089623792d+306) 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 = (1.0 / (b1 / a2)) * (a1 / b2);
	double tmp;
	if (t_0 <= -1.040164306635014e+306) {
		tmp = t_1;
	} else if (t_0 <= -7.4e-323) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 6.474204089623792e+306) {
		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 = (1.0 / (b1 / a2)) * (a1 / b2)
	tmp = 0
	if t_0 <= -1.040164306635014e+306:
		tmp = t_1
	elif t_0 <= -7.4e-323:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 6.474204089623792e+306:
		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(Float64(1.0 / Float64(b1 / a2)) * Float64(a1 / b2))
	tmp = 0.0
	if (t_0 <= -1.040164306635014e+306)
		tmp = t_1;
	elseif (t_0 <= -7.4e-323)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 6.474204089623792e+306)
		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 = (1.0 / (b1 / a2)) * (a1 / b2);
	tmp = 0.0;
	if (t_0 <= -1.040164306635014e+306)
		tmp = t_1;
	elseif (t_0 <= -7.4e-323)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 6.474204089623792e+306)
		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[(N[(1.0 / N[(b1 / a2), $MachinePrecision]), $MachinePrecision] * N[(a1 / b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.040164306635014e+306], t$95$1, If[LessEqual[t$95$0, -7.4e-323], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 6.474204089623792e+306], 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	2.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.040164306635014e306 or -7.41098e-323 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 6.4742040896237921e306 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 26.1

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

   2. Simplified 15.8

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

   3. Applied egg-rr 6.5

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

   4. Applied egg-rr 3.9

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}} \]

   if -1.040164306635014e306 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -7.41098e-323 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 6.4742040896237921e306

   1. Initial program 0.7

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

   2. Simplified 7.7

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

   3. Applied egg-rr 14.1

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

   4. Taylor expanded in a1 around 0 0.7

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.040164306635014 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -7.4 \cdot 10^{-323}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 6.474204089623792 \cdot 10^{+306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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