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

Average Error: 6.1 → 0.4

Time: 21.8s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{+262}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot \frac{2}{a}}{\frac{2}{y}}\\ \end{array} \]

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 -2e+226)
     (+ x (/ (- z t) (/ a y)))
     (if (<= t_1 1e+262)
       (+ x (/ t_1 a))
       (+ x (/ (* (- z t) (/ 2.0 a)) (/ 2.0 y)))))))

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 <= -2e+226) {
		tmp = x + ((z - t) / (a / y));
	} else if (t_1 <= 1e+262) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (((z - t) * (2.0 / a)) / (2.0 / y));
	}
	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 <= (-2d+226)) then
        tmp = x + ((z - t) / (a / y))
    else if (t_1 <= 1d+262) then
        tmp = x + (t_1 / a)
    else
        tmp = x + (((z - t) * (2.0d0 / a)) / (2.0d0 / y))
    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 <= -2e+226) {
		tmp = x + ((z - t) / (a / y));
	} else if (t_1 <= 1e+262) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (((z - t) * (2.0 / a)) / (2.0 / y));
	}
	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 <= -2e+226:
		tmp = x + ((z - t) / (a / y))
	elif t_1 <= 1e+262:
		tmp = x + (t_1 / a)
	else:
		tmp = x + (((z - t) * (2.0 / a)) / (2.0 / y))
	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 <= -2e+226)
		tmp = Float64(x + Float64(Float64(z - t) / Float64(a / y)));
	elseif (t_1 <= 1e+262)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(Float64(Float64(z - t) * Float64(2.0 / a)) / Float64(2.0 / y)));
	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 <= -2e+226)
		tmp = x + ((z - t) / (a / y));
	elseif (t_1 <= 1e+262)
		tmp = x + (t_1 / a);
	else
		tmp = x + (((z - t) * (2.0 / a)) / (2.0 / y));
	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, -2e+226], N[(x + N[(N[(z - t), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+262], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(z - t), $MachinePrecision] * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / N[(2.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.1
Target	0.6
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -1.99999999999999992e226

   1. Initial program 34.3

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

   2. Simplified 0.4

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

      [Start] 34.3	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 34.3	\[ x + \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.4	\[ x + \color{blue}{y \cdot \frac{z - t}{a}} \]

   3. Applied egg-rr 0.4

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

   if -1.99999999999999992e226 < (*.f64 y (-.f64 z t)) < 1e262

   1. Initial program 0.4

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

   if 1e262 < (*.f64 y (-.f64 z t))

   1. Initial program 43.0

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

   2. Simplified 0.4

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

      [Start] 43.0	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 43.0	\[ x + \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.4	\[ x + \color{blue}{y \cdot \frac{z - t}{a}} \]

   3. Applied egg-rr 0.4

      \[\leadsto x + \color{blue}{\frac{\left(z - t\right) \cdot \frac{2}{a}}{\frac{2}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 10^{+262}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot \frac{2}{a}}{\frac{2}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+217}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{+217}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Error	16.2
Cost	1108

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{a}\\ t_2 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-229}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-291}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	15.1
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{z - t}}\\ t_2 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-233}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	15.0
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{z - t}}\\ t_2 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;x \leq -7 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	14.5
Cost	1108

\[\begin{array}{l} t_1 := \frac{z - t}{\frac{a}{y}}\\ t_2 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;x \leq -8 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-225}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-300}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	28.8
Cost	1044

\[\begin{array}{l} t_1 := -\frac{t}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-227}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-303}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	18.2
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{a}{z}}\\ t_2 := -\frac{t}{\frac{a}{y}}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+217}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	7.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-14}:\\ \;\;\;\;t \cdot \frac{y}{-a} + x\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-245}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 10
Error	9.7
Cost	776

\[\begin{array}{l} t_1 := \left(-\frac{t}{\frac{a}{y}}\right) + x\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.85 \cdot 10^{-56}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	9.6
Cost	776

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-16}:\\ \;\;\;\;t \cdot \frac{y}{-a} + x\\ \mathbf{elif}\;t \leq 3.85 \cdot 10^{-56}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{t}{\frac{a}{y}}\right) + x\\ \end{array} \]

Alternative 12
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-86}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	28.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-30}:\\ \;\;\;\;-\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	28.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-30}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	28.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-180}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	27.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-180}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-136}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	27.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	2.5
Cost	576

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

Alternative 19
Error	31.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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)))