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

Average Error: 6.7 → 0.7

Time: 18.5s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a}\\ \mathbf{if}\;a \leq -2.95 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ x (* y (/ (- z t) a)))))
   (if (<= a -2.95e+27)
     t_1
     (if (<= a 1.15e-6) (+ x (/ (* y (- z t)) a)) 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 = x + (y * ((z - t) / a));
	double tmp;
	if (a <= -2.95e+27) {
		tmp = t_1;
	} else if (a <= 1.15e-6) {
		tmp = x + ((y * (z - t)) / a);
	} 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 = x + (y * ((z - t) / a))
    if (a <= (-2.95d+27)) then
        tmp = t_1
    else if (a <= 1.15d-6) then
        tmp = x + ((y * (z - t)) / a)
    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 = x + (y * ((z - t) / a));
	double tmp;
	if (a <= -2.95e+27) {
		tmp = t_1;
	} else if (a <= 1.15e-6) {
		tmp = x + ((y * (z - t)) / a);
	} 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 = x + (y * ((z - t) / a))
	tmp = 0
	if a <= -2.95e+27:
		tmp = t_1
	elif a <= 1.15e-6:
		tmp = x + ((y * (z - t)) / a)
	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(x + Float64(y * Float64(Float64(z - t) / a)))
	tmp = 0.0
	if (a <= -2.95e+27)
		tmp = t_1;
	elseif (a <= 1.15e-6)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	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 = x + (y * ((z - t) / a));
	tmp = 0.0;
	if (a <= -2.95e+27)
		tmp = t_1;
	elseif (a <= 1.15e-6)
		tmp = x + ((y * (z - t)) / a);
	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[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.95e+27], t$95$1, If[LessEqual[a, 1.15e-6], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	6.7
Target	0.7
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if a < -2.95000000000000022e27 or 1.15e-6 < a

   1. Initial program 10.3

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

   2. Simplified 0.4

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

      [Start] 10.3	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      rational.json-simplify-2 [=>] 10.3	\[ x + \frac{\color{blue}{\left(z - t\right) \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.4	\[ x + \color{blue}{y \cdot \frac{z - t}{a}} \]

   if -2.95000000000000022e27 < a < 1.15e-6

   1. Initial program 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.95 \cdot 10^{+27}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	3028

\[\begin{array}{l} t_1 := \frac{z - t}{\frac{a}{y}}\\ t_2 := \frac{y \cdot \left(z - t\right)}{a}\\ t_3 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{+42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-134}:\\ \;\;\;\;y \cdot \frac{z}{a} + x\\ \mathbf{elif}\;t_2 \leq 10^{+102}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	5.3
Cost	2640

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a}\\ t_2 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.9
Cost	1996

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 10^{+102}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 4
Error	34.6
Cost	1836

\[\begin{array}{l} t_1 := t \cdot \frac{y}{-a}\\ \mathbf{if}\;a \leq -2.05 \cdot 10^{+272}:\\ \;\;\;\;y \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.52 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-192}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-104}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-24}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+106}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	33.1
Cost	1704

\[\begin{array}{l} t_1 := t \cdot \frac{y}{-a}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -5.9 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -4.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+106}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	20.3
Cost	1108

\[\begin{array}{l} t_1 := y \cdot \frac{z}{a} + x\\ t_2 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -2.05 \cdot 10^{+272}:\\ \;\;\;\;y \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	1.9
Cost	1096

\[\begin{array}{l} t_1 := x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;z - t \leq -1 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 5 \cdot 10^{+35}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.6
Cost	976

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{a}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	19.1
Cost	976

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	17.1
Cost	976

\[\begin{array}{l} t_1 := y \cdot \frac{z}{a} + x\\ t_2 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -23500000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	10.0
Cost	776

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a} + x\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+56}:\\ \;\;\;\;\left(-y \cdot \frac{t}{a}\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	10.4
Cost	776

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right) + x\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+150}:\\ \;\;\;\;z \cdot \frac{y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	28.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-160}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	30.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(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)))