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

Average Error: 6.1 → 2.0

Time: 7.5s

Precision: binary64

Cost: 576

Math

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

↓

\[x + \frac{z - x}{\frac{t}{y}} \]

FPCore

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

↓

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

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 + ((z - x) / (t / y));
}

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 + ((z - x) / (t / y))
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 + ((z - x) / (t / y));
}

Python

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

↓

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

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(z - x) / Float64(t / y)))
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 + ((z - x) / (t / y));
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[(z - x), $MachinePrecision] / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	6.1
Target	2.1
Herbie	2.0

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

Derivation

1. Initial program 6.1

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

2. Simplified 2.1

   \[\leadsto \color{blue}{x + \frac{y}{t} \cdot \left(z - x\right)} \]
   Proof
   (+.f64 x (*.f64 (/.f64 y t) (-.f64 z x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y (-.f64 z x)) t))): 41 points increase in error, 23 points decrease in error

3. Applied egg-rr 2.0

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

4. Final simplification 2.0

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

Alternatives

Alternative 1
Error	18.9
Cost	712

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-54}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.1
Cost	712

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.2
Cost	712

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	26.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	26.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-54}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	26.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-54}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	2.1
Cost	576

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

Alternative 8
Error	31.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))