Quotient of products

Average Error: 11.0 → 5.5

Time: 5.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{a2}{b1} \cdot a1}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+273}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-266}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+285}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* (/ a2 b1) a1) b2)))
   (if (<= (* b1 b2) -1e+273)
     t_0
     (if (<= (* b1 b2) -5e-266)
       (* a2 (/ a1 (* b1 b2)))
       (if (<= (* b1 b2) 2e-181)
         t_0
         (if (<= (* b1 b2) 1e+285)
           (* a1 (/ a2 (* b1 b2)))
           (* (/ a2 b2) (/ a1 b1))))))))

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 = ((a2 / b1) * a1) / b2;
	double tmp;
	if ((b1 * b2) <= -1e+273) {
		tmp = t_0;
	} else if ((b1 * b2) <= -5e-266) {
		tmp = a2 * (a1 / (b1 * b2));
	} else if ((b1 * b2) <= 2e-181) {
		tmp = t_0;
	} else if ((b1 * b2) <= 1e+285) {
		tmp = a1 * (a2 / (b1 * b2));
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	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) :: tmp
    t_0 = ((a2 / b1) * a1) / b2
    if ((b1 * b2) <= (-1d+273)) then
        tmp = t_0
    else if ((b1 * b2) <= (-5d-266)) then
        tmp = a2 * (a1 / (b1 * b2))
    else if ((b1 * b2) <= 2d-181) then
        tmp = t_0
    else if ((b1 * b2) <= 1d+285) then
        tmp = a1 * (a2 / (b1 * b2))
    else
        tmp = (a2 / b2) * (a1 / b1)
    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 = ((a2 / b1) * a1) / b2;
	double tmp;
	if ((b1 * b2) <= -1e+273) {
		tmp = t_0;
	} else if ((b1 * b2) <= -5e-266) {
		tmp = a2 * (a1 / (b1 * b2));
	} else if ((b1 * b2) <= 2e-181) {
		tmp = t_0;
	} else if ((b1 * b2) <= 1e+285) {
		tmp = a1 * (a2 / (b1 * b2));
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	return tmp;
}

Python

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

↓

def code(a1, a2, b1, b2):
	t_0 = ((a2 / b1) * a1) / b2
	tmp = 0
	if (b1 * b2) <= -1e+273:
		tmp = t_0
	elif (b1 * b2) <= -5e-266:
		tmp = a2 * (a1 / (b1 * b2))
	elif (b1 * b2) <= 2e-181:
		tmp = t_0
	elif (b1 * b2) <= 1e+285:
		tmp = a1 * (a2 / (b1 * b2))
	else:
		tmp = (a2 / b2) * (a1 / b1)
	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(Float64(a2 / b1) * a1) / b2)
	tmp = 0.0
	if (Float64(b1 * b2) <= -1e+273)
		tmp = t_0;
	elseif (Float64(b1 * b2) <= -5e-266)
		tmp = Float64(a2 * Float64(a1 / Float64(b1 * b2)));
	elseif (Float64(b1 * b2) <= 2e-181)
		tmp = t_0;
	elseif (Float64(b1 * b2) <= 1e+285)
		tmp = Float64(a1 * Float64(a2 / Float64(b1 * b2)));
	else
		tmp = Float64(Float64(a2 / b2) * Float64(a1 / b1));
	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 = ((a2 / b1) * a1) / b2;
	tmp = 0.0;
	if ((b1 * b2) <= -1e+273)
		tmp = t_0;
	elseif ((b1 * b2) <= -5e-266)
		tmp = a2 * (a1 / (b1 * b2));
	elseif ((b1 * b2) <= 2e-181)
		tmp = t_0;
	elseif ((b1 * b2) <= 1e+285)
		tmp = a1 * (a2 / (b1 * b2));
	else
		tmp = (a2 / b2) * (a1 / b1);
	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[(N[(a2 / b1), $MachinePrecision] * a1), $MachinePrecision] / b2), $MachinePrecision]}, If[LessEqual[N[(b1 * b2), $MachinePrecision], -1e+273], t$95$0, If[LessEqual[N[(b1 * b2), $MachinePrecision], -5e-266], N[(a2 * N[(a1 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b1 * b2), $MachinePrecision], 2e-181], t$95$0, If[LessEqual[N[(b1 * b2), $MachinePrecision], 1e+285], N[(a1 * N[(a2 / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 / b2), $MachinePrecision] * N[(a1 / b1), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.0
Target	11.6
Herbie	5.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 b1 b2) < -9.99999999999999945e272 or -4.99999999999999992e-266 < (*.f64 b1 b2) < 2.00000000000000009e-181

   1. Initial program 27.2

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

   2. Simplified 28.8

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

   3. Applied egg-rr 7.4

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

   if -9.99999999999999945e272 < (*.f64 b1 b2) < -4.99999999999999992e-266

   1. Initial program 5.0

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

   2. Simplified 5.4

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

   3. Taylor expanded in a1 around 0 5.0

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

   4. Simplified 5.6

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

   if 2.00000000000000009e-181 < (*.f64 b1 b2) < 9.9999999999999998e284

   1. Initial program 5.0

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

   2. Simplified 5.1

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

   if 9.9999999999999998e284 < (*.f64 b1 b2)

   1. Initial program 19.8

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

   2. Simplified 19.4

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

   3. Applied egg-rr 3.1

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

   4. Taylor expanded in a2 around 0 7.6

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

   5. Applied egg-rr 1.7

      \[\leadsto \color{blue}{\frac{a2}{b2} \cdot \frac{a1}{b1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -1 \cdot 10^{+273}:\\ \;\;\;\;\frac{\frac{a2}{b1} \cdot a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -5 \cdot 10^{-266}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{a2}{b1} \cdot a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 10^{+285}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022325 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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