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

Average Error: 6.3 → 11.3

Time: 7.0s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{y}{a} + x\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2729909712006244 \cdot 10^{-13}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ (* t (/ y a)) x)))
   (if (<= t -5.5e+120)
     t_1
     (if (<= t 1.2729909712006244e-13) (- x (/ y (/ a z))) t_1))))

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 = (t * (y / a)) + x;
	double tmp;
	if (t <= -5.5e+120) {
		tmp = t_1;
	} else if (t <= 1.2729909712006244e-13) {
		tmp = x - (y / (a / z));
	} else {
		tmp = t_1;
	}
	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 = (t * (y / a)) + x
    if (t <= (-5.5d+120)) then
        tmp = t_1
    else if (t <= 1.2729909712006244d-13) then
        tmp = x - (y / (a / z))
    else
        tmp = t_1
    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 = (t * (y / a)) + x;
	double tmp;
	if (t <= -5.5e+120) {
		tmp = t_1;
	} else if (t <= 1.2729909712006244e-13) {
		tmp = x - (y / (a / z));
	} else {
		tmp = t_1;
	}
	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 = (t * (y / a)) + x
	tmp = 0
	if t <= -5.5e+120:
		tmp = t_1
	elif t <= 1.2729909712006244e-13:
		tmp = x - (y / (a / z))
	else:
		tmp = t_1
	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(t * Float64(y / a)) + x)
	tmp = 0.0
	if (t <= -5.5e+120)
		tmp = t_1;
	elseif (t <= 1.2729909712006244e-13)
		tmp = Float64(x - Float64(y / Float64(a / z)));
	else
		tmp = t_1;
	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 = (t * (y / a)) + x;
	tmp = 0.0;
	if (t <= -5.5e+120)
		tmp = t_1;
	elseif (t <= 1.2729909712006244e-13)
		tmp = x - (y / (a / z));
	else
		tmp = t_1;
	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[(t * N[(y / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -5.5e+120], t$95$1, If[LessEqual[t, 1.2729909712006244e-13], N[(x - N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	6.3
Target	0.7
Herbie	11.3

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

Derivation

1. Split input into 2 regimes

2. if t < -5.50000000000000003e120 or 1.27299097120062436e-13 < t

   1. Initial program 10.3

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

   2. Taylor expanded in z around 0 16.7

      \[\leadsto x - \color{blue}{-1 \cdot \frac{y \cdot t}{a}} \]

   3. Simplified 10.1

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

   4. Applied egg-rr 10.1

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

   if -5.50000000000000003e120 < t < 1.27299097120062436e-13

   1. Initial program 3.9

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

   2. Taylor expanded in z around 0 3.9

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

   3. Simplified 3.9

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

   4. Taylor expanded in z around inf 12.1

      \[\leadsto x - \frac{y}{\color{blue}{\frac{a}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+120}:\\ \;\;\;\;t \cdot \frac{y}{a} + x\\ \mathbf{elif}\;t \leq 1.2729909712006244 \cdot 10^{-13}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a} + x\\ \end{array} \]

Alternatives

Alternative 1
Error	20.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3903158634800512 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.251745803953851 \cdot 10^{-54}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	18.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5639119254976997 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 24151598768137.45:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	15.0
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{\frac{a}{z}}\\ \mathbf{if}\;x \leq -7.443501525911245 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.7413345596741496 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	28.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -7.731210893386766 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.251745803953851 \cdot 10^{-54}:\\ \;\;\;\;\frac{y \cdot z}{-a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7627249209107922 \cdot 10^{-149}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7413345596741496 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	31.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))