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

Average Error: 6.8 → 1.9

Time: 10.1s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, y, z, t)
	tmp = x + ((y / t) * (z - x));
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_] := N[(x + N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	6.8
Target	1.9
Herbie	1.9

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

Derivation

1. Initial program 6.8

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

2. Simplified 1.9

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

   [Start] 6.8	\[ x + \frac{y \cdot \left(z - x\right)}{t} \]
   associate-*l/ [<=] 1.9	\[ x + \color{blue}{\frac{y}{t} \cdot \left(z - x\right)} \]

3. Final simplification 1.9

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

Alternatives

Alternative 1
Error	23.4
Cost	1504

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \left(z - x\right)\\ t_2 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-251}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	22.4
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	26.9
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-148} \lor \neg \left(x \leq 2.5 \cdot 10^{-49}\right) \land x \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	26.2
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-146} \lor \neg \left(x \leq 7.5 \cdot 10^{-65}\right) \land x \leq 2.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	10.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-88}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+166}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \]

Alternative 6
Error	11.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-46} \lor \neg \left(t \leq 1.02 \cdot 10^{-140}\right):\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \]

Alternative 7
Error	31.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023054 
(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)))