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

Average Error: 6.2 → 0.8

Time: 1.6min

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+217}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a} \cdot y + x\\ \end{array} \]

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 -4e+217)
     (- x (* (/ y a) (- t z)))
     (if (<= t_1 5e+298) (+ x t_1) (+ (* (/ (- z t) a) y) x)))))

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 <= -4e+217) {
		tmp = x - ((y / a) * (t - z));
	} else if (t_1 <= 5e+298) {
		tmp = x + t_1;
	} else {
		tmp = (((z - t) / a) * y) + x;
	}
	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 <= (-4d+217)) then
        tmp = x - ((y / a) * (t - z))
    else if (t_1 <= 5d+298) then
        tmp = x + t_1
    else
        tmp = (((z - t) / a) * y) + x
    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 <= -4e+217) {
		tmp = x - ((y / a) * (t - z));
	} else if (t_1 <= 5e+298) {
		tmp = x + t_1;
	} else {
		tmp = (((z - t) / a) * y) + x;
	}
	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 <= -4e+217:
		tmp = x - ((y / a) * (t - z))
	elif t_1 <= 5e+298:
		tmp = x + t_1
	else:
		tmp = (((z - t) / a) * y) + x
	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 <= -4e+217)
		tmp = Float64(x - Float64(Float64(y / a) * Float64(t - z)));
	elseif (t_1 <= 5e+298)
		tmp = Float64(x + t_1);
	else
		tmp = Float64(Float64(Float64(Float64(z - t) / a) * y) + x);
	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 <= -4e+217)
		tmp = x - ((y / a) * (t - z));
	elseif (t_1 <= 5e+298)
		tmp = x + t_1;
	else
		tmp = (((z - t) / a) * y) + x;
	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, -4e+217], N[(x - N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], N[(x + t$95$1), $MachinePrecision], N[(N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	6.2
Target	0.7
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -3.99999999999999984e217

   1. Initial program 30.0

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

   2. Simplified 3.4

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

   3. Applied egg-rr 3.4

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

   if -3.99999999999999984e217 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.0000000000000003e298

   1. Initial program 0.4

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

   if 5.0000000000000003e298 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 58.3

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

   2. Simplified 1.0

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

   3. Applied egg-rr 3.0

      \[\leadsto \color{blue}{\frac{z - t}{a} \cdot y + x} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	29.3
Cost	1376

\[\begin{array}{l} t_1 := \frac{z}{a} \cdot y\\ t_2 := \frac{y}{-a} \cdot t\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-192}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	29.4
Cost	1376

\[\begin{array}{l} t_1 := \frac{z}{a} \cdot y\\ t_2 := \frac{y}{-a} \cdot t\\ \mathbf{if}\;x \leq -4 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-278}:\\ \;\;\;\;\frac{-t}{a} \cdot y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	2.5
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+255}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{a} \cdot y\\ \end{array} \]

Alternative 4
Error	0.6
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \frac{z - t}{a} \cdot y + x\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+185}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	29.3
Cost	1244

\[\begin{array}{l} t_1 := \frac{y}{-a} \cdot t\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-198}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-279}:\\ \;\;\;\;\frac{-y \cdot t}{a}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	15.9
Cost	976

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(z - t\right)\\ t_2 := \frac{y \cdot z}{a} + x\\ \mathbf{if}\;x \leq -0.0071:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-218}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	12.5
Cost	976

\[\begin{array}{l} t_1 := x - \frac{t}{a} \cdot y\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{y \cdot z}{a} + x\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	18.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+49}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-8}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-159}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	29.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-216}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	31.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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)))