Quotient of products

Average Error: 11.0 → 6.8

Time: 4.5s

Precision: binary64

Cost: 1488

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{if}\;a1 \cdot a2 \leq -2 \cdot 10^{+49}:\\ \;\;\;\;\frac{a2}{\frac{b2 \cdot b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -4 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-157}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (/ (* a1 a2) b1) b2)))
   (if (<= (* a1 a2) -2e+49)
     (/ a2 (/ (* b2 b1) a1))
     (if (<= (* a1 a2) -4e-183)
       t_0
       (if (<= (* a1 a2) 1e-157)
         (/ a2 (* b1 (/ b2 a1)))
         (if (<= (* a1 a2) 2e+92) t_0 (* a1 (/ (/ a2 b2) 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 = ((a1 * a2) / b1) / b2;
	double tmp;
	if ((a1 * a2) <= -2e+49) {
		tmp = a2 / ((b2 * b1) / a1);
	} else if ((a1 * a2) <= -4e-183) {
		tmp = t_0;
	} else if ((a1 * a2) <= 1e-157) {
		tmp = a2 / (b1 * (b2 / a1));
	} else if ((a1 * a2) <= 2e+92) {
		tmp = t_0;
	} else {
		tmp = a1 * ((a2 / b2) / 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 = ((a1 * a2) / b1) / b2
    if ((a1 * a2) <= (-2d+49)) then
        tmp = a2 / ((b2 * b1) / a1)
    else if ((a1 * a2) <= (-4d-183)) then
        tmp = t_0
    else if ((a1 * a2) <= 1d-157) then
        tmp = a2 / (b1 * (b2 / a1))
    else if ((a1 * a2) <= 2d+92) then
        tmp = t_0
    else
        tmp = a1 * ((a2 / b2) / 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 = ((a1 * a2) / b1) / b2;
	double tmp;
	if ((a1 * a2) <= -2e+49) {
		tmp = a2 / ((b2 * b1) / a1);
	} else if ((a1 * a2) <= -4e-183) {
		tmp = t_0;
	} else if ((a1 * a2) <= 1e-157) {
		tmp = a2 / (b1 * (b2 / a1));
	} else if ((a1 * a2) <= 2e+92) {
		tmp = t_0;
	} else {
		tmp = a1 * ((a2 / b2) / b1);
	}
	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 (a1 * a2) <= -2e+49:
		tmp = a2 / ((b2 * b1) / a1)
	elif (a1 * a2) <= -4e-183:
		tmp = t_0
	elif (a1 * a2) <= 1e-157:
		tmp = a2 / (b1 * (b2 / a1))
	elif (a1 * a2) <= 2e+92:
		tmp = t_0
	else:
		tmp = a1 * ((a2 / b2) / 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(a1 * a2) / b1) / b2)
	tmp = 0.0
	if (Float64(a1 * a2) <= -2e+49)
		tmp = Float64(a2 / Float64(Float64(b2 * b1) / a1));
	elseif (Float64(a1 * a2) <= -4e-183)
		tmp = t_0;
	elseif (Float64(a1 * a2) <= 1e-157)
		tmp = Float64(a2 / Float64(b1 * Float64(b2 / a1)));
	elseif (Float64(a1 * a2) <= 2e+92)
		tmp = t_0;
	else
		tmp = Float64(a1 * Float64(Float64(a2 / b2) / 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 = ((a1 * a2) / b1) / b2;
	tmp = 0.0;
	if ((a1 * a2) <= -2e+49)
		tmp = a2 / ((b2 * b1) / a1);
	elseif ((a1 * a2) <= -4e-183)
		tmp = t_0;
	elseif ((a1 * a2) <= 1e-157)
		tmp = a2 / (b1 * (b2 / a1));
	elseif ((a1 * a2) <= 2e+92)
		tmp = t_0;
	else
		tmp = a1 * ((a2 / b2) / 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[(a1 * a2), $MachinePrecision] / b1), $MachinePrecision] / b2), $MachinePrecision]}, If[LessEqual[N[(a1 * a2), $MachinePrecision], -2e+49], N[(a2 / N[(N[(b2 * b1), $MachinePrecision] / a1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a1 * a2), $MachinePrecision], -4e-183], t$95$0, If[LessEqual[N[(a1 * a2), $MachinePrecision], 1e-157], N[(a2 / N[(b1 * N[(b2 / a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a1 * a2), $MachinePrecision], 2e+92], t$95$0, N[(a1 * N[(N[(a2 / b2), $MachinePrecision] / b1), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.0
Target	11.1
Herbie	6.8

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 a1 a2) < -1.99999999999999989e49

   1. Initial program 17.3

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

   2. Taylor expanded in a1 around 0 17.3

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

   3. Simplified 14.6

      \[\leadsto \color{blue}{\frac{a2}{\frac{b2 \cdot b1}{a1}}} \]
      Proof
      (/.f64 a2 (/.f64 (*.f64 b2 b1) a1)): 0 points increase in error, 0 points decrease in error
      (/.f64 a2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 b1 b2)) a1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a2 a1) (*.f64 b1 b2))): 47 points increase in error, 36 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a1 a2)) (*.f64 b1 b2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a1 a2) (Rewrite=> *-commutative_binary64 (*.f64 b2 b1))): 0 points increase in error, 0 points decrease in error

   if -1.99999999999999989e49 < (*.f64 a1 a2) < -4.00000000000000002e-183 or 9.99999999999999943e-158 < (*.f64 a1 a2) < 2.0000000000000001e92

   1. Initial program 3.4

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

   2. Simplified 11.4

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 50 points increase in error, 41 points decrease in error

   3. Applied egg-rr 3.3

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

   if -4.00000000000000002e-183 < (*.f64 a1 a2) < 9.99999999999999943e-158

   1. Initial program 13.5

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

   2. Applied egg-rr 5.4

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

   3. Applied egg-rr 5.5

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

   if 2.0000000000000001e92 < (*.f64 a1 a2)

   1. Initial program 20.2

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

   2. Simplified 14.5

      \[\leadsto \color{blue}{a1 \cdot \frac{a2}{b1 \cdot b2}} \]
      Proof
      (*.f64 a1 (/.f64 a2 (*.f64 b1 b2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 50 points increase in error, 41 points decrease in error

   3. Taylor expanded in a1 around 0 20.2

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

   4. Simplified 11.9

      \[\leadsto \color{blue}{a1 \cdot \frac{\frac{a2}{b2}}{b1}} \]
      Proof
      (*.f64 a1 (/.f64 (/.f64 a2 b2) b1)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a1 (/.f64 a2 b2)) b1)): 35 points increase in error, 37 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 a1 b1) (/.f64 a2 b2))): 61 points increase in error, 42 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))): 57 points increase in error, 78 points decrease in error
      (/.f64 (*.f64 a1 a2) (Rewrite=> *-commutative_binary64 (*.f64 b2 b1))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -2 \cdot 10^{+49}:\\ \;\;\;\;\frac{a2}{\frac{b2 \cdot b1}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq -4 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 10^{-157}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.5
Cost	1488

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b2 \cdot b1}\\ t_1 := \frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{if}\;b2 \cdot b1 \leq -2 \cdot 10^{+221}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;b2 \cdot b1 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b2 \cdot b1 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	5.0
Cost	1488

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b2 \cdot b1}\\ t_1 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{if}\;b2 \cdot b1 \leq -2 \cdot 10^{+221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b2 \cdot b1 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq 5 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b2 \cdot b1 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.0
Cost	1488

\[\begin{array}{l} t_0 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{if}\;b2 \cdot b1 \leq -2 \cdot 10^{+221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\frac{a2}{\frac{b2 \cdot b1}{a1}}\\ \mathbf{elif}\;b2 \cdot b1 \leq 5 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;a2 \cdot \frac{a1}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	5.0
Cost	1488

\[\begin{array}{l} t_0 := \frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{if}\;b2 \cdot b1 \leq -1 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;b2 \cdot b1 \leq 5 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;a2 \cdot \frac{a1}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	8.1
Cost	1228

\[\begin{array}{l} t_0 := a1 \cdot \frac{a2}{b2 \cdot b1}\\ t_1 := a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{if}\;b2 \cdot b1 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b2 \cdot b1 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq 2 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	7.7
Cost	1228

\[\begin{array}{l} t_0 := a2 \cdot \frac{a1}{b2 \cdot b1}\\ t_1 := a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{if}\;b2 \cdot b1 \leq -2 \cdot 10^{+221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b2 \cdot b1 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b2 \cdot b1 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	11.3
Cost	448

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

Reproduce

herbie shell --seed 2022329 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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