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

Average Error: 6.3 → 3.0

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -9.793361331116636 \cdot 10^{+146}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;t \leq 1.1791014341897874 \cdot 10^{-293}:\\ \;\;\;\;x + \frac{y \cdot z - y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{t}} \cdot \frac{z - x}{\sqrt{t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9.793361331116636e+146)
   (+ x (* y (/ (- z x) t)))
   (if (<= t 1.1791014341897874e-293)
     (+ x (/ (- (* y z) (* y x)) t))
     (+ x (* (/ y (sqrt t)) (/ (- z x) (sqrt t)))))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / t);
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.793361331116636e+146) {
		tmp = x + (y * ((z - x) / t));
	} else if (t <= 1.1791014341897874e-293) {
		tmp = x + (((y * z) - (y * x)) / t);
	} else {
		tmp = x + ((y / sqrt(t)) * ((z - x) / sqrt(t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y * (z - x)) / t)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9.793361331116636d+146)) then
        tmp = x + (y * ((z - x) / t))
    else if (t <= 1.1791014341897874d-293) then
        tmp = x + (((y * z) - (y * x)) / t)
    else
        tmp = x + ((y / sqrt(t)) * ((z - x) / sqrt(t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / t);
}

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.793361331116636e+146) {
		tmp = x + (y * ((z - x) / t));
	} else if (t <= 1.1791014341897874e-293) {
		tmp = x + (((y * z) - (y * x)) / t);
	} else {
		tmp = x + ((y / Math.sqrt(t)) * ((z - x) / Math.sqrt(t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x + ((y * (z - x)) / t)

↓

def code(x, y, z, t):
	tmp = 0
	if t <= -9.793361331116636e+146:
		tmp = x + (y * ((z - x) / t))
	elif t <= 1.1791014341897874e-293:
		tmp = x + (((y * z) - (y * x)) / t)
	else:
		tmp = x + ((y / math.sqrt(t)) * ((z - x) / math.sqrt(t)))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * Float64(z - x)) / t))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9.793361331116636e+146)
		tmp = Float64(x + Float64(y * Float64(Float64(z - x) / t)));
	elseif (t <= 1.1791014341897874e-293)
		tmp = Float64(x + Float64(Float64(Float64(y * z) - Float64(y * x)) / t));
	else
		tmp = Float64(x + Float64(Float64(y / sqrt(t)) * Float64(Float64(z - x) / sqrt(t))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y * (z - x)) / t);
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9.793361331116636e+146)
		tmp = x + (y * ((z - x) / t));
	elseif (t <= 1.1791014341897874e-293)
		tmp = x + (((y * z) - (y * x)) / t);
	else
		tmp = x + ((y / sqrt(t)) * ((z - x) / sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[t, -9.793361331116636e+146], N[(x + N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1791014341897874e-293], N[(x + N[(N[(N[(y * z), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[(N[(z - x), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	1.9
Herbie	3.0

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

Derivation

1. Split input into 3 regimes

2. if t < -9.7933613311166362e146

   1. Initial program 11.2

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

   2. Simplified 1.6

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

   3. Applied fma-udef_binary64 1.6

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

   if -9.7933613311166362e146 < t < 1.1791014341897874e-293

   1. Initial program 3.3

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

   2. Simplified 9.3

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

   3. Applied fma-udef_binary64 9.3

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

   4. Taylor expanded in t around inf 3.3

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

   if 1.1791014341897874e-293 < t

   1. Initial program 6.6

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

   2. Simplified 6.1

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

   3. Applied fma-udef_binary64 6.1

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

   4. Applied add-sqr-sqrt_binary64 6.2

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

   5. Applied *-un-lft-identity_binary64 6.2

      \[\leadsto y \cdot \frac{\color{blue}{1 \cdot \left(z - x\right)}}{\sqrt{t} \cdot \sqrt{t}} + x \]

   6. Applied times-frac_binary64 6.2

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{\sqrt{t}} \cdot \frac{z - x}{\sqrt{t}}\right)} + x \]

   7. Applied associate-*r*_binary64 3.4

      \[\leadsto \color{blue}{\left(y \cdot \frac{1}{\sqrt{t}}\right) \cdot \frac{z - x}{\sqrt{t}}} + x \]

   8. Simplified 3.3

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.793361331116636 \cdot 10^{+146}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;t \leq 1.1791014341897874 \cdot 10^{-293}:\\ \;\;\;\;x + \frac{y \cdot z - y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt{t}} \cdot \frac{z - x}{\sqrt{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))