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

Average Error: 6.0 → 1.2

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -7.079263599664848 \cdot 10^{+215}:\\ \;\;\;\;x + {\left(\frac{\frac{a}{z - t}}{y}\right)}^{-1}\\ \mathbf{elif}\;t_1 \leq -2.333286477993578 \cdot 10^{-173}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

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 (<= t_1 -7.079263599664848e+215)
     (+ x (pow (/ (/ a (- z t)) y) -1.0))
     (if (<= t_1 -2.333286477993578e-173)
       (+ x (/ t_1 a))
       (+ x (* (- z t) (/ y 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 <= -7.079263599664848e+215) {
		tmp = x + pow(((a / (z - t)) / y), -1.0);
	} else if (t_1 <= -2.333286477993578e-173) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + ((z - t) * (y / 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 <= (-7.079263599664848d+215)) then
        tmp = x + (((a / (z - t)) / y) ** (-1.0d0))
    else if (t_1 <= (-2.333286477993578d-173)) then
        tmp = x + (t_1 / a)
    else
        tmp = x + ((z - t) * (y / 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 <= -7.079263599664848e+215) {
		tmp = x + Math.pow(((a / (z - t)) / y), -1.0);
	} else if (t_1 <= -2.333286477993578e-173) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + ((z - t) * (y / 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 <= -7.079263599664848e+215:
		tmp = x + math.pow(((a / (z - t)) / y), -1.0)
	elif t_1 <= -2.333286477993578e-173:
		tmp = x + (t_1 / a)
	else:
		tmp = x + ((z - t) * (y / 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 <= -7.079263599664848e+215)
		tmp = Float64(x + (Float64(Float64(a / Float64(z - t)) / y) ^ -1.0));
	elseif (t_1 <= -2.333286477993578e-173)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / 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 <= -7.079263599664848e+215)
		tmp = x + (((a / (z - t)) / y) ^ -1.0);
	elseif (t_1 <= -2.333286477993578e-173)
		tmp = x + (t_1 / a);
	else
		tmp = x + ((z - t) * (y / 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[LessEqual[t$95$1, -7.079263599664848e+215], N[(x + N[Power[N[(N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2.333286477993578e-173], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $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	6.0
Target	0.7
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -7.0792635996648483e215

   1. Initial program 29.6

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

   2. Applied egg-rr 0.6

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

   if -7.0792635996648483e215 < (*.f64 y (-.f64 z t)) < -2.3332864779935781e-173

   1. Initial program 0.1

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

   if -2.3332864779935781e-173 < (*.f64 y (-.f64 z t))

   1. Initial program 5.6

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

   2. Taylor expanded in y around 0 5.6

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

   3. Simplified 1.9

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -7.079263599664848 \cdot 10^{+215}:\\ \;\;\;\;x + {\left(\frac{\frac{a}{z - t}}{y}\right)}^{-1}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq -2.333286477993578 \cdot 10^{-173}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))