Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Average Error: 6.3 → 1.0

Time: 8.9s

Precision: binary64

Cost: 1608

Math

\[x + \frac{y \cdot \left(z - t\right)}{a} \]

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+281}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 100000000000:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (+ x (/ (- z t) (/ a y)))))
   (if (<= t_1 -1e+281) t_2 (if (<= t_1 100000000000.0) (+ t_1 x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / a);
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = x + ((z - t) / (a / y));
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = t_2;
	} else if (t_1 <= 100000000000.0) {
		tmp = t_1 + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * (z - t)) / a)
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = x + ((z - t) / (a / y))
    if (t_1 <= (-1d+281)) then
        tmp = t_2
    else if (t_1 <= 100000000000.0d0) then
        tmp = t_1 + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / a);
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = x + ((z - t) / (a / y));
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = t_2;
	} else if (t_1 <= 100000000000.0) {
		tmp = t_1 + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return x + ((y * (z - t)) / a)

↓

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = x + ((z - t) / (a / y))
	tmp = 0
	if t_1 <= -1e+281:
		tmp = t_2
	elif t_1 <= 100000000000.0:
		tmp = t_1 + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y * Float64(z - t)) / a))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(x + Float64(Float64(z - t) / Float64(a / y)))
	tmp = 0.0
	if (t_1 <= -1e+281)
		tmp = t_2;
	elseif (t_1 <= 100000000000.0)
		tmp = Float64(t_1 + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y * (z - t)) / a);
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = x + ((z - t) / (a / y));
	tmp = 0.0;
	if (t_1 <= -1e+281)
		tmp = t_2;
	elseif (t_1 <= 100000000000.0)
		tmp = t_1 + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(z - t), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+281], t$95$2, If[LessEqual[t$95$1, 100000000000.0], N[(t$95$1 + x), $MachinePrecision], t$95$2]]]]

Target

Original	6.3
Target	0.7
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1e281 or 1e11 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 19.4

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]

   2. Simplified 2.8

      \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(z - t\right)} \]
      Proof
      (+.f64 x (*.f64 (/.f64 y a) (-.f64 z t))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y (-.f64 z t)) a))): 37 points increase in error, 24 points decrease in error

   3. Applied egg-rr 2.6

      \[\leadsto x + \color{blue}{\frac{z - t}{\frac{a}{y}}} \]

   if -1e281 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e11

   1. Initial program 0.4

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1 \cdot 10^{+281}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 100000000000:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+221}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\frac{t_1}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	30.7
Cost	1308

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right)\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-232}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-257}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+206}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	1.5
Cost	1096

\[\begin{array}{l} t_1 := x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;z - t \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 2 \cdot 10^{+44}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	19.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	16.0
Cost	712

\[\begin{array}{l} t_1 := x + y \cdot \frac{z}{a}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+180}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 7
Error	11.5
Cost	712

\[\begin{array}{l} t_1 := x - \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;t \leq -9 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	29.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	28.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-58}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	2.4
Cost	576

\[x + \left(z - t\right) \cdot \frac{y}{a} \]

Alternative 11
Error	31.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))