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

Average Error: 9.46% → 1.58%

Time: 12.7s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-83} \lor \neg \left(y \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -6.8e-83) (not (<= y 5e-54)))
   (- x (/ y (/ a (- z t))))
   (- x (/ (* y (- z t)) a))))

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 tmp;
	if ((y <= -6.8e-83) || !(y <= 5e-54)) {
		tmp = x - (y / (a / (z - t)));
	} else {
		tmp = x - ((y * (z - t)) / a);
	}
	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) :: tmp
    if ((y <= (-6.8d-83)) .or. (.not. (y <= 5d-54))) then
        tmp = x - (y / (a / (z - t)))
    else
        tmp = x - ((y * (z - t)) / a)
    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 tmp;
	if ((y <= -6.8e-83) || !(y <= 5e-54)) {
		tmp = x - (y / (a / (z - t)));
	} else {
		tmp = x - ((y * (z - t)) / a);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -6.8e-83) or not (y <= 5e-54):
		tmp = x - (y / (a / (z - t)))
	else:
		tmp = x - ((y * (z - t)) / a)
	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)
	tmp = 0.0
	if ((y <= -6.8e-83) || !(y <= 5e-54))
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	else
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / a));
	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)
	tmp = 0.0;
	if ((y <= -6.8e-83) || ~((y <= 5e-54)))
		tmp = x - (y / (a / (z - t)));
	else
		tmp = x - ((y * (z - t)) / a);
	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_] := If[Or[LessEqual[y, -6.8e-83], N[Not[LessEqual[y, 5e-54]], $MachinePrecision]], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Target

Original	9.46%
Target	1.29%
Herbie	1.58%

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

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999995e-83 or 5.00000000000000015e-54 < y

   1. Initial program 17.62

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

   2. Simplified 2.22

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

      [Start] 17.62	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 2.22	\[ x - \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   if -6.7999999999999995e-83 < y < 5.00000000000000015e-54

   1. Initial program 0.9

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

4. Final simplification 1.58

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-83} \lor \neg \left(y \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	44.71%
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \frac{-y}{a}\\ t_2 := \frac{y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-212}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	45.12%
Cost	1176

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-212}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-187}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	44.75%
Cost	1045

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-212}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-296}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-176} \lor \neg \left(x \leq 7.5 \cdot 10^{-80}\right) \land x \leq 1.55 \cdot 10^{-32}:\\ \;\;\;\;\frac{z}{\frac{-a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	44.91%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-212}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-297}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-179}:\\ \;\;\;\;\frac{z}{\frac{-a}{y}}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	24.54%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-168} \lor \neg \left(x \leq 2.45 \cdot 10^{-177}\right) \land \left(x \leq 4.9 \cdot 10^{-82} \lor \neg \left(x \leq 4.8 \cdot 10^{-30}\right)\right):\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

Alternative 6
Error	33.27%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-92} \lor \neg \left(x \leq -2.5 \cdot 10^{-168}\right) \land x \leq 0.00047:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	32.75%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	44.72%
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-168}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-32}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	3.71%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-201} \lor \neg \left(z \leq 6 \cdot 10^{-99}\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 10
Error	19.62%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-99} \lor \neg \left(z \leq 8.5 \cdot 10^{-11}\right):\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 11
Error	15.92%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-99} \lor \neg \left(z \leq 7 \cdot 10^{-11}\right):\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 12
Error	45.25%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-212}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-173}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	45.17%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-213}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	3.77%
Cost	576

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

Alternative 15
Error	48.78%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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)))