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

Average Accuracy: 90.2% → 99.3%

Time: 16.6s

Precision: binary64

Cost: 1481

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+276} \lor \neg \left(t_1 \leq 10^{+190}\right):\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t - y \cdot z}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (or (<= t_1 -5e+276) (not (<= t_1 1e+190)))
     (- x (* y (/ (- z t) a)))
     (+ x (/ (- (* y t) (* y z)) 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 t_1 = y * (z - t);
	double tmp;
	if ((t_1 <= -5e+276) || !(t_1 <= 1e+190)) {
		tmp = x - (y * ((z - t) / a));
	} else {
		tmp = x + (((y * t) - (y * z)) / 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) :: t_1
    real(8) :: tmp
    t_1 = y * (z - t)
    if ((t_1 <= (-5d+276)) .or. (.not. (t_1 <= 1d+190))) then
        tmp = x - (y * ((z - t) / a))
    else
        tmp = x + (((y * t) - (y * z)) / 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 t_1 = y * (z - t);
	double tmp;
	if ((t_1 <= -5e+276) || !(t_1 <= 1e+190)) {
		tmp = x - (y * ((z - t) / a));
	} else {
		tmp = x + (((y * t) - (y * z)) / a);
	}
	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)
	tmp = 0
	if (t_1 <= -5e+276) or not (t_1 <= 1e+190):
		tmp = x - (y * ((z - t) / a))
	else:
		tmp = x + (((y * t) - (y * z)) / 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)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if ((t_1 <= -5e+276) || !(t_1 <= 1e+190))
		tmp = Float64(x - Float64(y * Float64(Float64(z - t) / a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(y * t) - Float64(y * z)) / 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)
	t_1 = y * (z - t);
	tmp = 0.0;
	if ((t_1 <= -5e+276) || ~((t_1 <= 1e+190)))
		tmp = x - (y * ((z - t) / a));
	else
		tmp = x + (((y * t) - (y * z)) / 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_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+276], N[Not[LessEqual[t$95$1, 1e+190]], $MachinePrecision]], N[(x - N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y * t), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.2%
Target	99.0%
Herbie	99.3%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -5.00000000000000001e276 or 1.0000000000000001e190 < (*.f64 y (-.f64 z t))

   1. Initial program 45.7%

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

   2. Simplified 98.9%

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

      [Start] 45.7	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*r/ [<=] 98.9	\[ x - \color{blue}{y \cdot \frac{z - t}{a}} \]

   if -5.00000000000000001e276 < (*.f64 y (-.f64 z t)) < 1.0000000000000001e190

   1. Initial program 99.4%

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

   2. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 99.4	\[ x - \frac{y \cdot \color{blue}{\left(z + \left(-t\right)\right)}}{a} \]
      distribute-rgt-in [=>] 99.4	\[ x - \frac{\color{blue}{z \cdot y + \left(-t\right) \cdot y}}{a} \]

   3. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ x - \frac{z \cdot y + \left(-t\right) \cdot y}{a} \]
      distribute-lft-neg-out [=>] 99.4	\[ x - \frac{z \cdot y + \color{blue}{\left(-t \cdot y\right)}}{a} \]
      unsub-neg [=>] 99.4	\[ x - \frac{\color{blue}{z \cdot y - t \cdot y}}{a} \]
      *-commutative [=>] 99.4	\[ x - \frac{z \cdot y - \color{blue}{y \cdot t}}{a} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{+276} \lor \neg \left(y \cdot \left(z - t\right) \leq 10^{+190}\right):\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t - y \cdot z}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+276} \lor \neg \left(t_1 \leq 10^{+190}\right):\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \end{array} \]

Alternative 2
Accuracy	53.4%
Cost	1244

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.72 \cdot 10^{-177}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	53.1%
Cost	1244

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-236}:\\ \;\;\;\;y \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	53.1%
Cost	1244

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-172}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	53.2%
Cost	1244

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 10^{-235}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	53.1%
Cost	1244

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-234}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{y \cdot z}{-a}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	96.9%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{+111} \lor \neg \left(z - t \leq 10^{-119}\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 8
Accuracy	96.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{+129} \lor \neg \left(z - t \leq 10^{-119}\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 9
Accuracy	90.0%
Cost	972

\[\begin{array}{l} t_1 := x - y \cdot \frac{z - t}{a}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-245}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-191}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	53.4%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-22} \lor \neg \left(x \leq -3.6 \cdot 10^{-86}\right) \land x \leq 8.2 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	70.9%
Cost	845

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+253} \lor \neg \left(z \leq 4.5 \cdot 10^{+271}\right):\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\frac{z}{a}\right)\\ \end{array} \]

Alternative 12
Accuracy	78.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-48} \lor \neg \left(t \leq 2.3 \cdot 10^{+121}\right):\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 13
Accuracy	80.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-24} \lor \neg \left(t \leq 1.9 \cdot 10^{+122}\right):\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 14
Accuracy	83.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-24} \lor \neg \left(t \leq 2.3 \cdot 10^{+121}\right):\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 15
Accuracy	53.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+168} \lor \neg \left(t \leq 2.9 \cdot 10^{+131}\right):\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	51.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(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)))