Quotient of products

Average Error: 11.2 → 2.9

Time: 3.8s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+291}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 10^{+294}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -2e+291)
     (/ (* a1 (/ a2 b1)) b2)
     (if (<= t_0 -5e-303)
       t_0
       (if (<= t_0 1e-288)
         (/ (* a1 (/ a2 b2)) b1)
         (if (<= t_0 1e+294) t_0 (* (/ 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 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+291) {
		tmp = (a1 * (a2 / b1)) / b2;
	} else if (t_0 <= -5e-303) {
		tmp = t_0;
	} else if (t_0 <= 1e-288) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if (t_0 <= 1e+294) {
		tmp = t_0;
	} 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 = (a1 * a2) / (b1 * b2)
    if (t_0 <= (-2d+291)) then
        tmp = (a1 * (a2 / b1)) / b2
    else if (t_0 <= (-5d-303)) then
        tmp = t_0
    else if (t_0 <= 1d-288) then
        tmp = (a1 * (a2 / b2)) / b1
    else if (t_0 <= 1d+294) then
        tmp = t_0
    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 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -2e+291) {
		tmp = (a1 * (a2 / b1)) / b2;
	} else if (t_0 <= -5e-303) {
		tmp = t_0;
	} else if (t_0 <= 1e-288) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if (t_0 <= 1e+294) {
		tmp = t_0;
	} 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 = (a1 * a2) / (b1 * b2)
	tmp = 0
	if t_0 <= -2e+291:
		tmp = (a1 * (a2 / b1)) / b2
	elif t_0 <= -5e-303:
		tmp = t_0
	elif t_0 <= 1e-288:
		tmp = (a1 * (a2 / b2)) / b1
	elif t_0 <= 1e+294:
		tmp = t_0
	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(a1 * a2) / Float64(b1 * b2))
	tmp = 0.0
	if (t_0 <= -2e+291)
		tmp = Float64(Float64(a1 * Float64(a2 / b1)) / b2);
	elseif (t_0 <= -5e-303)
		tmp = t_0;
	elseif (t_0 <= 1e-288)
		tmp = Float64(Float64(a1 * Float64(a2 / b2)) / b1);
	elseif (t_0 <= 1e+294)
		tmp = t_0;
	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 = (a1 * a2) / (b1 * b2);
	tmp = 0.0;
	if (t_0 <= -2e+291)
		tmp = (a1 * (a2 / b1)) / b2;
	elseif (t_0 <= -5e-303)
		tmp = t_0;
	elseif (t_0 <= 1e-288)
		tmp = (a1 * (a2 / b2)) / b1;
	elseif (t_0 <= 1e+294)
		tmp = t_0;
	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[(a1 * a2), $MachinePrecision] / N[(b1 * b2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+291], N[(N[(a1 * N[(a2 / b1), $MachinePrecision]), $MachinePrecision] / b2), $MachinePrecision], If[LessEqual[t$95$0, -5e-303], t$95$0, If[LessEqual[t$95$0, 1e-288], N[(N[(a1 * N[(a2 / b2), $MachinePrecision]), $MachinePrecision] / b1), $MachinePrecision], If[LessEqual[t$95$0, 1e+294], t$95$0, N[(N[(a2 / b2), $MachinePrecision] * N[(a1 / b1), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	11.2
Target	11.2
Herbie	2.9

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9999999999999999e291

   1. Initial program 55.0

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

   2. Simplified 30.9

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

   3. Applied egg-rr 17.3

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

   if -1.9999999999999999e291 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.9999999999999998e-303 or 1.00000000000000006e-288 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.00000000000000007e294

   1. Initial program 0.8

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

   if -4.9999999999999998e-303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.00000000000000006e-288

   1. Initial program 13.1

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

   2. Simplified 7.8

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

   3. Applied egg-rr 3.9

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

   if 1.00000000000000007e294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 59.7

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

   2. Simplified 7.5

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{+291}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	2514

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq -2 \cdot 10^{-313} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+294}\right):\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.2
Cost	2513

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+291}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-313} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+294}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Alternative 3
Error	5.1
Cost	1490

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5 \cdot 10^{+165} \lor \neg \left(b1 \cdot b2 \leq -1 \cdot 10^{-279} \lor \neg \left(b1 \cdot b2 \leq 10^{-306}\right) \land b1 \cdot b2 \leq 5 \cdot 10^{+295}\right):\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array} \]

Alternative 4
Error	11.1
Cost	448

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

Reproduce

herbie shell --seed 2022340 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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