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

Average Error: 6.7 → 0.9

Time: 17.6s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+291}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+69}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

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 -1e+291)
     (- x (* y (/ (- z t) a)))
     (if (<= t_1 5e+69) (- 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 <= -1e+291) {
		tmp = x - (y * ((z - t) / a));
	} else if (t_1 <= 5e+69) {
		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 <= (-1d+291)) then
        tmp = x - (y * ((z - t) / a))
    else if (t_1 <= 5d+69) 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 <= -1e+291) {
		tmp = x - (y * ((z - t) / a));
	} else if (t_1 <= 5e+69) {
		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 <= -1e+291:
		tmp = x - (y * ((z - t) / a))
	elif t_1 <= 5e+69:
		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 <= -1e+291)
		tmp = Float64(x - Float64(y * Float64(Float64(z - t) / a)));
	elseif (t_1 <= 5e+69)
		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 <= -1e+291)
		tmp = x - (y * ((z - t) / a));
	elseif (t_1 <= 5e+69)
		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, -1e+291], N[(x - N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+69], N[(x - N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.7
Target	0.8
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -9.9999999999999996e290

   1. Initial program 55.7

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

   2. Simplified 0.2

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

      [Start] 55.7	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 55.7	\[ x - \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.2	\[ x - \color{blue}{y \cdot \frac{z - t}{a}} \]

   if -9.9999999999999996e290 < (*.f64 y (-.f64 z t)) < 5.00000000000000036e69

   1. Initial program 0.6

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

   if 5.00000000000000036e69 < (*.f64 y (-.f64 z t))

   1. Initial program 14.0

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

   2. Simplified 2.1

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

      [Start] 14.0	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-49 [=>] 2.1	\[ x - \color{blue}{\left(z - t\right) \cdot \frac{y}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1 \cdot 10^{+291}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 5 \cdot 10^{+69}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.4
Cost	1096

\[\begin{array}{l} t_1 := x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;z - t \leq -1 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq -2 \cdot 10^{-175}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.7
Cost	976

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+136}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -35:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	10.6
Cost	976

\[\begin{array}{l} t_1 := t \cdot \frac{y}{a} + x\\ t_2 := x - z \cdot \frac{y}{a}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-55}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{elif}\;z \leq 1.14 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	31.0
Cost	912

\[\begin{array}{l} t_1 := z \cdot \frac{-y}{a}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-256}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-284}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	31.1
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+89}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-256}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-286}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	5.7
Cost	840

\[\begin{array}{l} t_1 := x - y \cdot \frac{z - t}{a}\\ \mathbf{if}\;a \leq -2.05 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	27.8
Cost	780

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-124}:\\ \;\;\;\;-y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	15.7
Cost	712

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;t \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	9.8
Cost	712

\[\begin{array}{l} t_1 := x - z \cdot \frac{y}{a}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.05 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -6.3 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023073 
(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)))