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

Average Error: 6.1 → 0.3

Time: 23.2s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \left(\frac{t}{a} - \frac{z}{a}\right) \cdot y + x\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+261}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+233}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (+ (* (- (/ t a) (/ z a)) y) x)))
   (if (<= t_1 -2e+261) t_2 (if (<= t_1 2e+233) (- x (/ t_1 a)) 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);
	double t_2 = (((t / a) - (z / a)) * y) + x;
	double tmp;
	if (t_1 <= -2e+261) {
		tmp = t_2;
	} else if (t_1 <= 2e+233) {
		tmp = x - (t_1 / a);
	} 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)
    t_2 = (((t / a) - (z / a)) * y) + x
    if (t_1 <= (-2d+261)) then
        tmp = t_2
    else if (t_1 <= 2d+233) then
        tmp = x - (t_1 / a)
    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);
	double t_2 = (((t / a) - (z / a)) * y) + x;
	double tmp;
	if (t_1 <= -2e+261) {
		tmp = t_2;
	} else if (t_1 <= 2e+233) {
		tmp = x - (t_1 / a);
	} 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)
	t_2 = (((t / a) - (z / a)) * y) + x
	tmp = 0
	if t_1 <= -2e+261:
		tmp = t_2
	elif t_1 <= 2e+233:
		tmp = x - (t_1 / a)
	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(y * Float64(z - t))
	t_2 = Float64(Float64(Float64(Float64(t / a) - Float64(z / a)) * y) + x)
	tmp = 0.0
	if (t_1 <= -2e+261)
		tmp = t_2;
	elseif (t_1 <= 2e+233)
		tmp = Float64(x - Float64(t_1 / a));
	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);
	t_2 = (((t / a) - (z / a)) * y) + x;
	tmp = 0.0;
	if (t_1 <= -2e+261)
		tmp = t_2;
	elseif (t_1 <= 2e+233)
		tmp = x - (t_1 / a);
	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[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t / a), $MachinePrecision] - N[(z / a), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+261], t$95$2, If[LessEqual[t$95$1, 2e+233], N[(x - N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Target

Original	6.1
Target	0.6
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -1.9999999999999999e261 or 1.99999999999999995e233 < (*.f64 y (-.f64 z t))

   1. Initial program 38.5

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

   2. Taylor expanded in y around 0 0.6

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

   if -1.9999999999999999e261 < (*.f64 y (-.f64 z t)) < 1.99999999999999995e233

   1. Initial program 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2 \cdot 10^{+261}:\\ \;\;\;\;\left(\frac{t}{a} - \frac{z}{a}\right) \cdot y + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+233}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} - \frac{z}{a}\right) \cdot y + x\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	2640

\[\begin{array}{l} t_1 := \left(\frac{t}{a} - \frac{z}{a}\right) \cdot y\\ t_2 := \frac{y \cdot \left(z - t\right)}{a}\\ t_3 := -t_2\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -50000000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+179}:\\ \;\;\;\;\frac{y \cdot t}{a} + x\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.8
Cost	2576

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := -t_1\\ t_3 := \frac{t}{a} \cdot y + x\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+302}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -50000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+179}:\\ \;\;\;\;\frac{y \cdot t}{a} + x\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	1.8
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \left(\frac{t}{a} - \frac{z}{a}\right) \cdot y\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+261}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	13.5
Cost	1240

\[\begin{array}{l} t_1 := \frac{t}{a} \cdot y + x\\ t_2 := x - \frac{y \cdot z}{a}\\ t_3 := \frac{y \cdot t}{a} + x\\ \mathbf{if}\;z \leq -0.00116:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-230}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+162}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	16.5
Cost	976

\[\begin{array}{l} t_1 := \frac{y \cdot t}{a} + x\\ t_2 := \frac{t}{a} \cdot y + x\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.26 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;-\frac{y \cdot z}{a}\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	28.3
Cost	780

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-236}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-127}:\\ \;\;\;\;-\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	19.2
Cost	712

\[\begin{array}{l} t_1 := \frac{t}{a} \cdot y + x\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-214}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -600:\\ \;\;\;\;\frac{y \cdot t}{a} + x\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-32}:\\ \;\;\;\;x - \frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \end{array} \]

Alternative 9
Error	28.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))