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

Average Accuracy: 90.4% → 99.3%

Time: 12.8s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+268}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+181}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

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 (<= t_1 -5e+268)
     (+ x (* y (/ (- z t) a)))
     (if (<= t_1 4e+181) (+ x (/ t_1 a)) (+ x (/ y (/ a (- z t))))))))

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+268) {
		tmp = x + (y * ((z - t) / a));
	} else if (t_1 <= 4e+181) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	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+268)) then
        tmp = x + (y * ((z - t) / a))
    else if (t_1 <= 4d+181) then
        tmp = x + (t_1 / a)
    else
        tmp = x + (y / (a / (z - t)))
    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+268) {
		tmp = x + (y * ((z - t) / a));
	} else if (t_1 <= 4e+181) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	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+268:
		tmp = x + (y * ((z - t) / a))
	elif t_1 <= 4e+181:
		tmp = x + (t_1 / a)
	else:
		tmp = x + (y / (a / (z - t)))
	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+268)
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / a)));
	elseif (t_1 <= 4e+181)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	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+268)
		tmp = x + (y * ((z - t) / a));
	elseif (t_1 <= 4e+181)
		tmp = x + (t_1 / a);
	else
		tmp = x + (y / (a / (z - t)));
	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[LessEqual[t$95$1, -5e+268], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+181], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	90.4%
Target	98.9%
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 3 regimes

2. if (*.f64 y (-.f64 z t)) < -5.0000000000000002e268

   1. Initial program 29.8%

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

   2. Simplified 99.7%

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

      [Start] 29.8	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 29.8	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 99.7	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   3. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ \mathsf{fma}\left(y, \frac{z - t}{a}, x\right) \]
      fma-udef [=>] 99.7	\[ \color{blue}{y \cdot \frac{z - t}{a} + x} \]

   if -5.0000000000000002e268 < (*.f64 y (-.f64 z t)) < 3.9999999999999997e181

   1. Initial program 99.4%

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

   if 3.9999999999999997e181 < (*.f64 y (-.f64 z t))

   1. Initial program 60.6%

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

   2. Simplified 98.4%

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

      [Start] 60.6	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 98.4	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{+268}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 4 \cdot 10^{+181}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+298} \lor \neg \left(t_1 \leq 4 \cdot 10^{+181}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \end{array} \]

Alternative 2
Accuracy	55.6%
Cost	1176

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-131}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	54.9%
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{y}{a}\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-130}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	55.1%
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{y}{a}\right)\\ t_2 := z \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-252}:\\ \;\;\;\;\frac{y \cdot t}{-a}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	97.5%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -1 \cdot 10^{-67} \lor \neg \left(z - t \leq 5 \cdot 10^{-45}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 6
Accuracy	53.4%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-250} \lor \neg \left(x \leq 8.5 \cdot 10^{-229}\right) \land x \leq 7 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	53.7%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-249}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	75.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-143} \lor \neg \left(x \leq 1.7 \cdot 10^{+21}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 9
Accuracy	84.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-18} \lor \neg \left(z \leq 33000000000000\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t}}\\ \end{array} \]

Alternative 10
Accuracy	68.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	96.1%
Cost	576

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

Alternative 12
Accuracy	51.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023136 
(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)))