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

Average Accuracy: 90.9% → 97.4%

Time: 15.4s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+137}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-109}:\\ \;\;\;\;x - t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \end{array} \]

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)))
   (if (<= t_1 -5e+137)
     (- x (* (- z t) (/ y a)))
     (if (<= t_1 2e-109) (- x t_1) (+ x (/ (- t z) (/ a y)))))))

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 tmp;
	if (t_1 <= -5e+137) {
		tmp = x - ((z - t) * (y / a));
	} else if (t_1 <= 2e-109) {
		tmp = x - t_1;
	} else {
		tmp = x + ((t - z) / (a / y));
	}
	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) :: tmp
    t_1 = (y * (z - t)) / a
    if (t_1 <= (-5d+137)) then
        tmp = x - ((z - t) * (y / a))
    else if (t_1 <= 2d-109) then
        tmp = x - t_1
    else
        tmp = x + ((t - z) / (a / y))
    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 tmp;
	if (t_1 <= -5e+137) {
		tmp = x - ((z - t) * (y / a));
	} else if (t_1 <= 2e-109) {
		tmp = x - t_1;
	} else {
		tmp = x + ((t - z) / (a / y));
	}
	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
	tmp = 0
	if t_1 <= -5e+137:
		tmp = x - ((z - t) * (y / a))
	elif t_1 <= 2e-109:
		tmp = x - t_1
	else:
		tmp = x + ((t - z) / (a / y))
	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)
	tmp = 0.0
	if (t_1 <= -5e+137)
		tmp = Float64(x - Float64(Float64(z - t) * Float64(y / a)));
	elseif (t_1 <= 2e-109)
		tmp = Float64(x - t_1);
	else
		tmp = Float64(x + Float64(Float64(t - z) / Float64(a / y)));
	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;
	tmp = 0.0;
	if (t_1 <= -5e+137)
		tmp = x - ((z - t) * (y / a));
	elseif (t_1 <= 2e-109)
		tmp = x - t_1;
	else
		tmp = x + ((t - z) / (a / y));
	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]}, If[LessEqual[t$95$1, -5e+137], N[(x - N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-109], N[(x - t$95$1), $MachinePrecision], N[(x + N[(N[(t - z), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	90.9%
Target	99.0%
Herbie	97.4%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.0000000000000002e137

   1. Initial program 70.6%

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

   2. Simplified 94.2%

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

      [Start] 70.6	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 94.2	\[ x - \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   if -5.0000000000000002e137 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e-109

   1. Initial program 99.2%

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

   if 2e-109 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 87.0%

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

   2. Simplified 96.2%

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

      [Start] 87.0	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 96.2	\[ x - \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   3. Applied egg-rr 96.2%

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

      [Start] 96.2	\[ x - \frac{y}{a} \cdot \left(z - t\right) \]
      associate-*l/ [=>] 87.0	\[ x - \color{blue}{\frac{y \cdot \left(z - t\right)}{a}} \]
      *-commutative [=>] 87.0	\[ x - \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      associate-/l* [=>] 96.2	\[ x - \color{blue}{\frac{z - t}{\frac{a}{y}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -5 \cdot 10^{+137}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{-109}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.4%
Cost	1609

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+137} \lor \neg \left(t_1 \leq 2 \cdot 10^{-109}\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t_1\\ \end{array} \]

Alternative 2
Accuracy	50.2%
Cost	1572

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{+250}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+134}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -120:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{z}{\frac{-a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	65.6%
Cost	1504

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ t_2 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+287}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;z \leq -5.3 \cdot 10^{+250}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1400:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-113}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	51.4%
Cost	1308

\[\begin{array}{l} t_1 := z \cdot \frac{-y}{a}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1920:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	50.7%
Cost	1308

\[\begin{array}{l} t_1 := z \cdot \frac{-y}{a}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	50.4%
Cost	1308

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+107}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{z}{\frac{-a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	50.9%
Cost	1308

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+133}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7.5:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+64}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+107}:\\ \;\;\;\;\frac{z}{\frac{-a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	69.3%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-71} \lor \neg \left(x \leq 4.5 \cdot 10^{+17}\right) \land x \leq 5.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	65.4%
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+96}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	79.3%
Cost	844

\[\begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -0.185:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-80}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	56.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-219}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	56.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-225}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	96.5%
Cost	576

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

Alternative 14
Accuracy	52.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(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)))