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

Average Error: 5.7 → 0.5

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -8.432815853165983 \cdot 10^{+146}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t_1 \leq 5.196317462097143 \cdot 10^{+225}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t - z}{a}}{{y}^{-1}}\\ \end{array} \]

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 -8.432815853165983e+146)
     (+ x (* (/ y a) (- t z)))
     (if (<= t_1 5.196317462097143e+225)
       (+ x (/ (* y (- t z)) a))
       (+ x (/ (/ (- t z) a) (pow y -1.0)))))))

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 <= -8.432815853165983e+146) {
		tmp = x + ((y / a) * (t - z));
	} else if (t_1 <= 5.196317462097143e+225) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = x + (((t - z) / a) / pow(y, -1.0));
	}
	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 <= (-8.432815853165983d+146)) then
        tmp = x + ((y / a) * (t - z))
    else if (t_1 <= 5.196317462097143d+225) then
        tmp = x + ((y * (t - z)) / a)
    else
        tmp = x + (((t - z) / a) / (y ** (-1.0d0)))
    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 <= -8.432815853165983e+146) {
		tmp = x + ((y / a) * (t - z));
	} else if (t_1 <= 5.196317462097143e+225) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = x + (((t - z) / a) / Math.pow(y, -1.0));
	}
	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 <= -8.432815853165983e+146:
		tmp = x + ((y / a) * (t - z))
	elif t_1 <= 5.196317462097143e+225:
		tmp = x + ((y * (t - z)) / a)
	else:
		tmp = x + (((t - z) / a) / math.pow(y, -1.0))
	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 <= -8.432815853165983e+146)
		tmp = Float64(x + Float64(Float64(y / a) * Float64(t - z)));
	elseif (t_1 <= 5.196317462097143e+225)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = Float64(x + Float64(Float64(Float64(t - z) / a) / (y ^ -1.0)));
	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 <= -8.432815853165983e+146)
		tmp = x + ((y / a) * (t - z));
	elseif (t_1 <= 5.196317462097143e+225)
		tmp = x + ((y * (t - z)) / a);
	else
		tmp = x + (((t - z) / a) / (y ^ -1.0));
	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, -8.432815853165983e+146], N[(x + N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5.196317462097143e+225], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] / N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.7
Target	0.7
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -8.4328158531659829e146

   1. Initial program 20.7

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

   2. Taylor expanded in y around 0 20.7

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

   3. Simplified 0.9

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

   if -8.4328158531659829e146 < (*.f64 y (-.f64 z t)) < 5.1963174620971431e225

   1. Initial program 0.4

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

   if 5.1963174620971431e225 < (*.f64 y (-.f64 z t))

   1. Initial program 32.3

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

   2. Applied egg-rr 1.0

      \[\leadsto x - \color{blue}{{\left(\frac{\frac{a}{y}}{z - t}\right)}^{-1}} \]

   3. Applied egg-rr 1.0

      \[\leadsto x - \color{blue}{\frac{\frac{z - t}{a}}{{y}^{-1}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -8.432815853165983 \cdot 10^{+146}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 5.196317462097143 \cdot 10^{+225}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t - z}{a}}{{y}^{-1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))